**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
import java.util.*;
import java.io.*;
public class E {
  public static void main(String[] args) throws IOException {
    BufferedReader r = new BufferedReader(new InputStreamReader(System.in));
		PrintWriter pw = new PrintWriter(System.out);

    StringTokenizer st = new StringTokenizer(r.readLine());
    int T = Integer.parseInt(st.nextToken());
    long mod = 1000000007l;
    for(int i = 0; i < T; i++) {
      st = new StringTokenizer(r.readLine());
      int n = Integer.parseInt(st.nextToken());
      int k = Integer.parseInt(st.nextToken());
      long[] value = new long[n];
      st = new StringTokenizer(r.readLine());
      for(int j = 0; j < n; j++) {
        value[j] = Long.parseLong(st.nextToken());
      }
      int reg = n - k;
      if(reg == 0) {
        long alice = 0l;
        for(int j = 0; j < n; j++) {
          alice += value[j];
          alice %= mod;
        }
        pw.println(alice + "  " + 0);
      } else {
        long special = 0l;
        for(int j = 0; j < k; j++) {
          special += value[j];
          special %= mod;
        }
        special *= inv((long) k, mod);
        special %= mod;
        long regular = 0l;
        for(int j = k; j < n; j++) {
          regular += value[j];
          regular %= mod;
        }
        regular *= inv((long) (n - k), mod);
        regular %= mod;
        long alice = 0l;
        long bob = 0l;
        if(reg % 2 == 1) {
          alice += ((inv(2l, mod)*special)*(k)) % mod;
          bob += ((inv(2l, mod)*special)*(k)) % mod;
          alice += (regular * ((reg + 1)/2))%mod;
          bob += (regular * ((reg - 1)/2))%mod;
          alice%=mod;
          bob%=mod;
        } else {
          alice += ((inv(2l, mod)*regular)*(n - k)) % mod;
          bob += ((inv(2l, mod)*regular)*(n - k)) % mod;
          alice += ((((reg/2 + 1)*special)%mod * inv((long) (reg + 1), mod))%mod * k)%mod;
          bob += ((((reg/2)*special)%mod * inv((long) (reg + 1), mod))%mod * k)%mod;
          alice%=mod;
          bob%=mod;
        }
        pw.println(alice + " " + bob);
      }
    }
    pw.close();
  }

  public static long inv(long x, long mod) {
    long target = mod - 2;
    int[] bin = new int[30];
    int index = 29;
    while(target > 0) {
      bin[index] = (int) target%2;
      target/=2;
      index--;
    }
    index = 0;
    long ret = 1;
    while(index < 30) {
      ret *= ret;
      ret %= mod;
      if(bin[index] == 1) {
        ret *= x;
        ret %= mod;
      }
      index++;
    }
    return ret;
  }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1
4 2
1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
const int MOD = 1e9+7;
const int nx = 8e5+10;
long long f[nx];
long long binp(long long a, long long n){
	long long r = 1;
	while(n){
		if(n&1) r = r*a%MOD;
		a = a*a%MOD;
		n>>=1;
	}
	return r;
}
long long inv(long long x){
	return binp(x, MOD-2);
}
long long choose(int n, int k){
	if(k > n || k < 0) return 0;
	return (f[n]*inv(f[n-k])%MOD)*inv(f[k])%MOD;
}
long long sb(int nboxes, int nitems){
	return choose(nitems+nboxes-1, nitems);
}

void solve(){
	int n, k;cin >> n >> k;
	long long a[n], s = 0, sp = 0;for(int i = 0;i < n;i++) {
		cin >> a[i];
		s += a[i];
		if(i<k) sp+=a[i];
		s%=MOD;
		sp%=MOD;
	}
	if(k==n){
		cout << s << " 0\n";
		return;
	}
	int m = n-k;
	long long tk = f[n], sk = m*f[n-1]%MOD;
	long long rk = (tk-sk+MOD)%MOD;
	long long al = 0, np = s-sp;
	if(m&1){
		al = np*(inv(2)+inv(2*m))%MOD;
		long long g = 0;
		for(int i = 1;i <= k;i++){
			g += (i*((f[k]*f[n-i-1]%MOD)*inv(f[k-i])%MOD)%MOD)*m%MOD;
			g%=MOD;
		}
		g = (g*inv(f[n]*k%MOD))%MOD;
		g = (1-g+MOD)%MOD;
		al += (g*(inv(2)+inv(2*m))%MOD)*sp%MOD;
		al%=MOD;
	} else {
		al = inv(2)*np%MOD;
		long long g = 0;
		for(int i = 1;i <= k;i++){
			g += (i*((f[k]*f[n-i-1]%MOD)*inv(f[k-i])%MOD)%MOD)*m%MOD;
			g%=MOD;
		}
		g = (g*inv(f[n]*k%MOD))%MOD;
		al += (g+(1-g+MOD)*inv(2)%MOD)*sp%MOD;
		al%=MOD;
	}
	cout << al << ' ' << (s-al+MOD)%MOD << '\n';
}
int main(){
	f[0]=1;
	for(int i = 1;i < nx;i++) f[i]=f[i-1]*i%MOD;
	ios_base::sync_with_stdio(false);
	cin.tie(0);
	int t;cin >> t;
	while(t--) solve();
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1
3 3
1 2 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define int long long
const int mod=1e9+7;
const int maxn=4e5+10;
int t,n,k,a[maxn];
inline int qpow(int a,int b)
{
	int res=1;
	while(b)
	{
		if(b&1)res=res*a%mod;
		a=a*a%mod;
		b>>=1;
	}
	return res;
}
inline int frac(int a,int b){return a*qpow(b,mod-2)%mod;}
inline int calc(int n,int k)
{
	if(k==0)return 0;
	if((n-k)&1)return frac(k,2);
	return frac(k+(n-k)/2*k,1+(n-k)/2*2);
}
inline int calc2(int k){return (k+1)/2;}
signed main()
{
	scanf("%lld",&t);
	while(t--)
	{
		scanf("%lld%lld",&n,&k);
		for(int i=1;i<=n;i++)scanf("%lld",&a[i]);
		int sum=0,ans=0;
		for(int i=1;i<=k;i++)sum=(sum+a[i])%mod;
		ans=(ans+sum*frac(calc(n,k),k))%mod;sum=0;
		for(int i=k+1;i<=n;i++)sum=(sum+a[i])%mod;
		if(n!=k)ans=(ans+sum*frac(calc2(n-k),n-k))%mod;
		sum=0;
		for(int i=1;i<=n;i++)sum=(sum+a[i])%mod;
		printf("%lld %lld\n",(ans%mod+mod)%mod,((sum-ans)%mod+mod)%mod);
	}
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 1
1
2 2
1 1
3 1
1 1 1
4 2
1 1 1 1
5 2
1 1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
#author: sushmanth
from sys import stdin , stdout
input = stdin.readline

inp = lambda : list(map(int , input().split()))

"""
let,
x be the number of non special balls
y be the number of special balls
s1 be the sum of non special balls
s2 be the sum of special balls

=> turn flips are fixed dude
_ a _ b _ a _ b _ a _ b _ a _

if(x & 1)
    alice turns => (x + 1) / 2
    bob turns => (x + 1) / 2
else
    alice turns => (x / 2) + 1
    bob turns => x / 2

For alice,
choices of a non special ball
alice_choices = ncr(x - 1 , (x + 1) / 2 - 1)
special_balls_partitioning = ncr(x + y , x)
ways = alice_choices * special_balls_partitioning * fact[y] * fact[(x + 1) / 2] * fact[x / 2]
expected_value_of_alice = s1 * (ways / fact[n])

choices of a special ball
let i be the choices of alice
alice_choices = ncr(y - 1 , i - 1)
special_balls_partitioning = alice_choices * ncr(i + alice turns - 1 , alice_turns - 1) * ncr((y - i) + bob turns - 1 , bob_turns - 1)
ways = alice_choices * special_balls_partitioning * fact[i] * fact[y - i] * fact[x]
expected_value_of_alice += s2 * (ways / fact[n])

For Bob
(s1 + s2) - expected_value_of_alice

"""

mod = (10 ** 9) + 7
def add(a , b):return ((a%mod) + (b%mod))%mod
def sub(a , b):return (a - b + mod)%mod
def mul(a , b):return ((a%mod) * (b%mod))%mod
def ncr(n , r):
    if(n < r):return 0
    return mul(fact[n] , mul(facti[n - r] , facti[r]))
def div(a , b):return mul(a , pow(b , mod - 2 , mod))

fact , facti = [1] , [1]
size = 2 * (10**5) + 1
for i in range(1 , size):
    fact.append(mul(fact[-1] , i))
    facti.append(pow(fact[-1] , mod - 2 , mod))

def answer():

    s1 , s2 = 0 , 0
    for i in range(n):
        if(i < k):
            s2 += a[i]
        else:
            s1 += a[i]

    if(n == k):
        return [(s1 + s2) % mod , 0]

    x = n - k
    y = k
    if(x & 1):
        alice_turns = (x + 1) // 2
        bob_turns = (x + 1) // 2
    else:
        alice_turns = (x // 2) + 1
        bob_turns = (x // 2)

    #not special balls
    alice_choices = ncr(x - 1 , (x + 1) // 2 - 1)
    special_balls_partitioning = ncr(x + y , x)
    ways = alice_choices * special_balls_partitioning * fact[y] * fact[(x + 1) // 2] * fact[x // 2]
    ways %= mod
    expected_value_of_alice = s1 * div(ways , fact[n])
    expected_value_of_alice %= mod

    #special balls
    for i in range(1 , y + 1):
        alice_choices = ncr(y - 1 , i - 1)
        special_balls_partitioning = ncr(i + alice_turns - 1 , alice_turns - 1) * ncr((y - i) + bob_turns - 1 , bob_turns - 1)
        ways = alice_choices * special_balls_partitioning * fact[i] * fact[y - i] * fact[x]
        ways %= mod
        expected_value_of_alice += s2 * div(ways , fact[n])
        expected_value_of_alice %= mod

    expected_value_of_bob = (s1 + s2) - expected_value_of_alice
    expected_value_of_bob %= mod

    return [expected_value_of_alice , expected_value_of_bob]

for T in range(int(input())):

    n , k = inp()
    a = inp()

    print(*answer())

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 1
1
2 1
1 2
3 2
1 2 3
4 2
1 2 3 4
5 3
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
import java.io.BufferedOutputStream;
import java.io.BufferedReader;
import java.io.BufferedWriter;
import java.io.File;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.io.OutputStream;
import java.io.OutputStreamWriter;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.Scanner;
import java.util.Set;
import java.util.Stack;
import java.util.StringTokenizer;
import java.util.TreeSet;

public class Main {
	static kattio sc = new kattio();
	static PrintWriter out = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));

	public static void main(String[] args) {

		int t = sc.nextint();
		while (t-- > 0) {

			sovle();
		}

		out.close();

	}
	static void sovle() {
		n=sc.nextint();
		m=sc.nextint();
		int []arr=new int[n];
		long sum1=0;long sum2=0;
		for (int i = 0; i < arr.length; i++) {
			arr[i]=sc.nextint();
			if (i<m) {
				sum1+=arr[i];sum1%=mod;
			}else {
				sum2+=arr[i];sum2%=mod;
			}
		}
		long alis=((n-m)/2+1)*m%mod*sum1%mod*pow((n-m+1)*m%mod, mod-2)%mod+((n-m+1)/2)*sum2%mod*pow(n-m, mod-2)%mod;alis%=mod;
		long bob=(m+mod-((n-m)/2+1)*m%mod*pow(n-m+1, mod-2)%mod)*sum1%mod*pow(m, mod-2)%mod+(n-m-(n-m+1)/2+mod)%mod*sum2%mod*pow(n-m, mod-2)%mod;bob%=mod;
		out.println(alis+" "+bob);
	}
	static int n,m;
	static long [][]q;
	static long tot;
	static void sovle2() {
		n=sc.nextint();
		long k=sc.nextlong();
		int [][]arr=new int[n+1][2];arr[0][0]=-1;
		for (int i = 1; i < arr.length; i++) {
			arr[i][0]=sc.nextint();
			arr[i][1]=i;
		}
		Arrays.sort(arr,(a,b)->a[0]-b[0]);
		Queue<int[]>queue=new PriorityQueue<int[]>((a,b)->a[0]-b[0]);
		for (int i = 2; i < arr.length; i++) {
			int[]add=new int[3];
			add[0]=arr[i][0]^arr[i-1][0];
			add[1]=Math.min(arr[i][1], arr[i-1][1]);
			add[2]=Math.max(arr[i][1], arr[i-1][1]);
			queue.add(add);
		}
		int l=0;int r=n+1;
		TreeSet<Integer>lset=new TreeSet<Integer>();lset.add(0);
		TreeSet<Integer>rset=new TreeSet<Integer>();
		while (!queue.isEmpty()) {
			int[] x=queue.poll();
			if (x[1]>l&&x[2]<r) {
				long va=1l*(x[1]-l)*(r-x[2]);
				l=x[1];r=x[2];
				if (k>va) {
					k-=va;
				}else {
					out.println(x[0]);return;
				}
			}
			System.out.println(Arrays.toString(x)+" "+l+" "+r);
		}
	}

	static Map<Integer, Integer> lMap, rMap;
	static Map<Integer, TreeSet<Integer>> vMap;

	static int[] dx = { 0, 0, 1, -1 };
	static int[] dy = { 1, -1, 0, 0 };
	static Map<Integer, List<Integer>> lpMap;

	static long pow(long x, long a) {
		if (a == 0) {
			return 1;
		} else {
			long c = pow(x, a / 2);
			if (a % 2 == 0) {
				return c * c % mod;
			} else {
				return c * c % mod * x % mod;
			}
		}
	}

	static long mod = (long) (1e9+7);

	static long gcd(long a, long b) {
		return b == 0 ? a : gcd(b, a % b);
	}

	static int[][] c;

	static int lowbit(int x) {
		return x & -x;
	}

	static void add(int x, int v,int i) {
		while (x < c.length) {
			c[x][i] += v;

			x += lowbit(x);
		}
	}

	static long sum(int x,int i) {
		int ans = 0;
		while (x > 0) {
			// System.out.println(x);
			ans += c[x][i];
			x -= lowbit(x);
		}
		return ans;
	}

	static Map<Integer, List<Integer>> map;

	static void mapinit(int chu, int n) {
		map = new HashMap<Integer, List<Integer>>();
		for (int i = chu; i < n + chu; i++) {
			map.put(i, new ArrayList<Integer>());
		}
	}
}

class kattio extends PrintWriter {

	BufferedReader r;
	StringTokenizer st;

	public kattio() {
		this(System.in, System.out);
	}

	public kattio(InputStream i, OutputStream o) {
		super(o);
		r = new BufferedReader(new InputStreamReader(i));
	}

	public kattio(String in, String out) throws IOException {
		super(out);
		r = new BufferedReader(new FileReader(in));
	}

	public String next() {
		try {
			while (st == null || !st.hasMoreTokens()) {
				st = new StringTokenizer(r.readLine());
			}
			return st.nextToken();
		} catch (Exception e) {
			// TODO: handle exception
			return null;
		}
	}

	public int nextint() {
		return Integer.parseInt(next());
	}

	public long nextlong() {
		return Long.parseLong(next());
	}

	public double nextdouble() {
		return Double.parseDouble(next());
	}
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that can cause the algorithm to fail. Here are some considerations:

1. **Edge Cases:**
   - **Single Ball:** Test cases with only one ball.
   - **All Special Balls:** Test cases where all balls are special.
   - **No Special Balls:** Test cases where no balls are special.
   - **Equal Number of Special and Non-Special Balls:** Test cases where the number of special and non-special balls is equal.

2. **Large Values:**
   - Test cases with large values for `n` and `k` to ensure the algorithm handles large inputs correctly.

3. **Modulo Operations:**
   - Test cases that involve large sums and modulo operations to ensure the algorithm handles large intermediate results correctly.

4. **Turns and Special Ball Effects:**
   - Test cases where the number of turns and the effects of special balls are critical to the expected score.

Given these considerations, a small-scale test input that can expose potential issues in the buggy code is:

```plaintext
5
1 1
1000000
2 2
1000000 2000000
3 1
1000000 2000000 3000000
4 2
1000000 2000000 3000000 4000000
5 3
1000000 2000000 3000000 4000000 5000000
```

This test input includes:
- A case with a single ball (all special).
- A case with two special balls.
- A case with one special ball and two non-special balls.
- A case with two special balls and two non-special balls.
- A case with three special balls and two non-special balls.

These cases should help expose any issues with the algorithm's handling of edge cases and large values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
MOD = 10**9 + 7

def mod_inverse(a, p):
    return pow(a, p - 2, p)

def solve():
    t = int(input())
    results = []
    for _ in range(t):
        n, k = map(int, input().split())
        l = list(map(int, input().split()))

        special_balls = l[:k]
        non_special_balls = l[k:]

        total_sum = sum(l)

        # Total number of non-special balls
        nonSpeCount = n - k

        if nonSpeCount == 0:
            results.append((total_sum, 0))
            continue

        avgSpe = sum(special_balls)
        avgNonSpe = sum(non_special_balls)

        avgNonSpe_div = avgNonSpe * mod_inverse(nonSpeCount, MOD) % MOD

        a = avgNonSpe_div * ((nonSpeCount + 1) // 2) % MOD
        b = avgNonSpe_div * (nonSpeCount // 2) % MOD

        spBlock = nonSpeCount + 1
        spCountA = (spBlock + 1) // 2
        spCountB = spBlock // 2

        avgSpe_div = avgSpe * mod_inverse(spBlock, MOD) % MOD

        a = (a + avgSpe_div * spCountA % MOD) % MOD
        b = (b + avgSpe_div * spCountB % MOD) % MOD

        results.append((a, b))

    for res in results:
        print(res[0], res[1])

if __name__ == "__main__":
    solve()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1
4 3
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: I Love Balls
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alice and Bob are playing a game. There are $n$ balls, out of which $k$ are special. Each ball has a value associated with it.

The players play turn by turn. In each turn, the player randomly picks a ball and adds the value of the ball to their score, which is $0$ at the beginning of the game. The selected ball is removed from the game. If the ball was special, the same player takes the next turn if at least one ball is remaining. If the ball picked was not special, the next player plays their turn.

They play this game until no balls are remaining in the game. Alice plays first.

Find the expected score that both the players have at the end of the game modulo $10^9+7$.

Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

Input: There are multiple test cases. The first line of the input contains an integer $t$, the number of test cases ($1 \le t \le 2 \cdot 10^5$).

Each test case description is on a new line. The first line of the test case contains two integers $n$ and $k$ in the respective order separated by a space ($1 \le k \le n \le 4 \cdot 10^5$).

The second line of the test case contains $n$ integers: $v_1, v_2, \ldots, v_n$, the value for each ball separated by spaces. The first $k$ balls are special ($1 \le v_i \le 10^7$).

The sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output: Output two integers per test case in a new line, the expected score of Alice and the expected score of Bob modulo $10^9+7$.

Examples:
input:
1
5 2
10 20 5 15 25
output:
45 30

input:
5
1 1
732507
2 2
5817860 5398510
5 1
2122894 4951549 2750585 7821535 3214167
8 4
1405323 5069867 6883092 6972029 328406 2478975 7628890 9973340
4 2
9662050 3566134 3996473 9872255
output:
732507 0
11216370 0
810642660 210218077
722402997 318336932
349086489 678010430

input:
5
3 3
1095611 8219204 7773462
2 1
8176490 2774103
3 1
9178636 5138057 3367761
12 9
7597698 6843019 2298534 1522386 4969588 1340345 3967362 9152890 6689668 9986080 4745473 7407325
10 5
6986368 2397882 5804127 6980694 3740836 3215836 5195724 3179261 4136769 4544231
output:
17088277 0
6862348 4088245
677038671 340645790
36949997 29570371
725118051 321063684

Note: In the first test case, Alice's expected score is $45$, and Bob's is $30$ at the end of the game.

Buggy Code:
import java.io.PrintStream;
import java.util.Scanner;

public class E_956 {
    static int mod = 1000000007;

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        PrintStream out = new PrintStream(System.out);

        while (t-- > 0) {
            int n = sc.nextInt();
            int k = sc.nextInt();

            long[] a = new long[n];
            for (int i = 0; i < n; i++) {
                a[i] = sc.nextLong();
            }

            long s1 = 0;
            long s2 = 0;
            for (int i = 0; i < n; i++) {
                if (i < k) {
                    s1 = (s1 + a[i]) % mod;
                } else {
                    s2 = (s2 + a[i]) % mod;
                }
            }
            long total = (s1 + s2) % mod;

            int x = ((n - k) / 2) + 1;

            long alicescore =(((long) x * fpow(n - k + 1, mod - 2) % mod * s1) % mod
                    + (((long) (n - k + 1) / 2) * s2 % mod * fpow(n - k, mod - 2)) % mod) % mod;

            long bobscore = (total - alicescore + mod) % mod;

            out.println(alicescore + " " + bobscore);
        }
    }

    private static long fpow(int a, int m) {
        if (m == 0) {
            return 1;
        }
        long hp = fpow(a, m / 2);
        long ans = (hp * hp) % mod;
        if (m % 2 == 0) {
            return ans;
        }
        return (ans * a) % mod;
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 1
10000000
2 1
1 10000000
3 2
10000000 1 1
4 2
10000000 10000000 1 1
5 3
10000000 10000000 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
#include <iostream>
#include <vector>
#include <algorithm>

const int MOD = 998244353;

int main() {
    int n;
    std::cin >> n;

    std::vector<int> a(n), b(n), c(n);
    for (int i = 0; i < n; ++i) {
        std::cin >> a[i];
    }
    for (int i = 0; i < n; ++i) {
        std::cin >> b[i];
    }
    for (int i = 0; i < n; ++i) {
        std::cin >> c[i];
    }

    std::vector<std::vector<long long>> dp(n + 1, std::vector<long long>(n + 1, 0));
    dp[0][0] = 1;

    for (int i = 1; i <= n; ++i) {
        for (int x = 0; x < i; ++x) {
            for (int y = 0; y <= x; ++y) {
                if (dp[i - 1][x] == 0) continue;

                int z = i - 1 - x - y;
                if (z > y) continue;

                long long cost = (long long)a[x] * b[y] % MOD * c[z] % MOD;
                dp[i][x + 1] = (dp[i][x + 1] + dp[i - 1][x]) % MOD;
                dp[i][x] = (dp[i][x] + dp[i - 1][x] * (long long)(n - i)) % MOD;
                dp[i][x] = (dp[i][x] + dp[i - 1][x] * (long long)y) % MOD;
                dp[i][x] = (dp[i][x] - dp[i - 1][x] * (long long)z + MOD) % MOD;
            }
        }
    }

    for (int i = 1; i <= n; ++i) {
        std::cout << dp[i][i] << " ";
    }
    std::cout << std::endl;

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider the following aspects:

1. **Edge Cases**: Inputs that are at the boundaries of the problem constraints, such as the smallest and largest possible values for \( n \).
2. **Special Cases**: Inputs that might cause the algorithm to behave unexpectedly, such as arrays with repeated values, zeros, or very large numbers.
3. **Complexity**: Inputs that maximize the number of operations and interactions between the different elements of the arrays.

Given the constraints \( 1 \le n \le 700 \), we will choose a small-scale test input that is complete and valid. Let's focus on a case where \( n \) is relatively small but still complex enough to potentially expose the bug.

### Chosen Test Input
- **\( n = 4 \)**: This is a small but non-trivial size that can still capture the essence of the problem.
- **Cost Arrays**: We will use arrays with a mix of small and large values to ensure that the multiplication and modulo operations are exercised.

Here is the test input:

```plaintext
4
1 2 3 4
4 3 2 1
1 2 3 4
```

### Explanation
- **Array \( a \)**: [1, 2, 3, 4]
  - This array is increasing, which means the prefix maximums will be at positions 1, 2, 3, and 4.
- **Array \( b \)**: [4, 3, 2, 1]
  - This array is decreasing, which means the suffix maximums will be at positions 4, 3, 2, and 1.
- **Array \( c \)**: [1, 2, 3, 4]
  - This array is increasing, which means the ascents will be at positions 2, 3, and 4.

This input is designed to:
- Ensure that the prefix and suffix maximums are well-defined and spread across the array.
- Ensure that the number of ascents is maximized, which will test the cost calculation thoroughly.
- Use a mix of small and large values to ensure that the modulo operation is tested.

This test input should help in exposing any issues in the dynamic programming logic of the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
const int N=701,mod=998244353;
int a[1<<20],b[1<<20],rev[1<<20];
int n,tot=1,ans[N],f[N][N],g[N][N];
int A[N],B[N],C[N],fact[N],ifact[N];

int read() {
	int x=0; char c=0; while (!isdigit(c) ) c=getchar();
	while (isdigit(c) ) x=(x<<3)+(x<<1)+(c&15),c=getchar();
	return x;
}
int power(int a,int b)
{
	int res=1;
	while (b) {
		if (b&1) res=1ll*res*a%mod;
		a=1ll*a*a%mod,b>>=1;
	}
	return res;
}
void NTT(int *a)
{
	for (int i=0;i<tot;i++)
	if (i<rev[i]) swap(a[i],a[rev[i] ]);
	for (int len=1;len<tot;len<<=1)
	{
		int g=power(3,(mod-1)/(len<<1) );
		for (int i=0;i<tot;i+=len<<1)
			for (int j=0,gk=1;j<len;j++,gk=1ll*gk*g%mod) {
				int x=a[i|j],y=1ll*a[i|len|j]*gk%mod;
				a[i|j]=(x+y)%mod,a[i|len|j]=(x-y+mod)%mod;
			}
	}
}
void INTT(int *a) {
	NTT(a),reverse(a+1,a+tot); int inv=power(tot,mod-2);
	for (int i=0;i<tot;i++) a[i]=1ll*a[i]*inv%mod;
}
int main()
{
	n=read(),fact[0]=ifact[0]=ifact[1]=1;
	for (int i=1;i<=n;i++) fact[i]=1ll*fact[i-1]*i%mod;
	for (int i=2;i<=n;i++) ifact[i]=1ll*ifact[mod%i]*(mod-mod/i)%mod;
	for (int i=1;i<=n;i++) ifact[i]=1ll*ifact[i-1]*ifact[i]%mod;
	for (int i=1;i<=n;i++) A[i]=read();
	for (int i=1;i<=n;i++) B[i]=read();
	for (int i=0;i<n;i++) C[i]=read();
	f[1][0]=1,b[0]=B[1];
	ans[0]=1ll*A[1]*B[1]%mod*C[0]%mod;
	for (int i=1;i<n;i++)
	{
		memcpy(g,f,sizeof g);
		memset(f,0,sizeof f);
		for (int j=1;j<=i;j++)
			for (int k=0;k<i;k++) if (g[j][k]) {
				f[j+1][k+1]=(f[j+1][k+1]+g[j][k])%mod;
				f[j][k]=(f[j][k]+1ll*(k+1)*g[j][k]%mod)%mod;
				f[j][k+1]=(f[j][k+1]+1ll*(i-k-1)*g[j][k]%mod)%mod;
				a[n*i+k]=(a[n*i+k]+1ll*ifact[i]*g[j][k]%mod*A[j+1]%mod)%mod;
				if (i+1<n) b[n*i+i-k-1]=(b[n*i+i-k-1]+1ll*ifact[i]*g[j][k]%mod*B[j+1]%mod)%mod;
				ans[i]=(ans[i]+1ll*ifact[i]*g[j][k]%mod*A[1]%mod*B[j+1]%mod*C[i]%mod)%mod;
			}
	}
	int l=-1; while (tot<2*n*n) tot<<=1,l++;
	for (int i=0;i<tot;i++) rev[i]=rev[i>>1]>>1|(i&1)<<l;
	NTT(a),NTT(b);
	for (int i=0;i<tot;i++) a[i]=1ll*a[i]*b[i]%mod;
	INTT(a);
	for (int i=0;i<n;i++) {
		for (int j=0;j<i;j++)
		ans[i]=(ans[i]+1ll*a[n*i+j]*C[j+1]%mod)%mod;
		printf("%d ",1ll*ans[i]*fact[i]%mod);
	}
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 0 0 0 0
0 0 0 0 1
1 1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
import sys
import math
import sys
import sys
from math import log2, ceil

MOD = 998244353

def read_input():
    n = int(sys.stdin.readline())
    a = list(map(int, sys.stdin.readline().split()))
    b = list(map(int, sys.stdin.readline().split()))
    c = list(map(int, sys.stdin.readline().split()))
    return n, a, b, c

def prepare_factorials(n):
    fact = [1] * (n+1)
    for i in range(1, n+1):
        fact[i] = fact[i-1] * i % MOD
    return fact

def compute_f(n, a, b, c, fact):
    f = [0] * (n+1)
    f[1] = a[0] * b[0] * c[0] % MOD
    for k in range(2, n+1):
        f[k] = fact[k]
    # Placeholder for the actual computation
    # Detailed implementation is required based on combinatorial insights
    return f[1:n+1]

def main():
    n, a, b, c = read_input()
    fact = prepare_factorials(n)
    result = compute_f(n, a, b, c, fact)
    print(' '.join(map(str, result)))

if __name__ == "__main__":
    main()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider edge cases and scenarios that are likely to highlight issues in the implementation. Here are the steps to think through this:

1. **Edge Cases for `n`**:
   - Small values of `n` (e.g., 1, 2) to ensure basic functionality.
   - Medium values of `n` (e.g., 10, 50) to test more complex permutations.
   - Large values of `n` (e.g., 700) to test performance and correctness under the upper limit.

2. **Special Values for Cost Arrays**:
   - Zero values in the cost arrays to test edge cases where multiplication results in zero.
   - Large values in the cost arrays to test the modulo operation.
   - Non-uniform values to test the combinatorial logic.

3. **Combination of Edge Cases**:
   - Test with a mix of zero and large values in the cost arrays to ensure the algorithm handles mixed inputs correctly.

Given these considerations, let's create a small-scale test input that is complete and valid:

```plaintext
5
1 0 2 3 4
2 3 1 0 4
300000000 100000000 500000000 400000000 200000000
```

This test input includes:
- `n = 5` to test a medium-sized input.
- `a` array with a zero value to test edge cases.
- `b` array with a zero value to test edge cases.
- `c` array with large values to test the modulo operation.

This input should help expose any issues in the combinatorial logic and edge case handling in the provided buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
MOD = 998244353

def solve(n, a, b, c):
    # Initialize dp table
    dp = [[[0]*(n+1) for _ in range(n+1)] for __ in range(n+1)]

    # Base case initialization
    dp[1][1][0] = a[0]*b[0]

    # Fill the dp table
    for i in range(2, n+1):  # Length of current sequence
        for prefix_max in range(1, i+1):
            for ascents in range(i-1):
                # Case 1: New number is placed at position 0 (new prefix max)
                dp[i][prefix_max][ascents+1] += dp[i-1][prefix_max-1][ascents] * c[ascents]

                # Case 2: New number is not a new prefix max but could increase ascents/suffix max
                for j in range(prefix_max):
                    dp[i][prefix_max][ascents+1 if j > 0 else ascents] += dp[i-1][prefix_max][j] * c[j]

        # Normalize all entries in dp[i][][] under MOD
        for prefix_max in range(1, i+1):
            for ascents in range(i):
                dp[i][prefix_max][ascents] %= MOD

    # Compute f(n) from dp[n][][]
    result = [0] * (n + 1)
    for i in range(1, n+1):
        for prefix_max in range(1, i+1):
            for ascents in range(i):
                result[i] += dp[i][prefix_max][ascents] * b[prefix_max-1] % MOD
                result[i] %= MOD

    return ' '.join(map(str, result[1:]))

# Example usage:
n = int(input())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
c = list(map(int, input().split()))

print(solve(n, a, b, c))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases or scenarios that can maximize the failure probability in the faulty solution. Here are the steps to think through this:

1. **Edge Cases**:
   - **Single Element Array**: This is the simplest case and should be handled correctly.
   - **Two Element Array**: This is a minimal non-trivial case that can expose issues in handling prefix and suffix maximums.
   - **Three Element Array**: This is a more complex case that can expose issues in handling multiple ascents and maximums.

2. **Specific Scenarios**:
   - **All Elements Equal**: This can expose issues in handling multiple prefix and suffix maximums.
   - **Decreasing Sequence**: This can expose issues in handling suffix maximums.
   - **Increasing Sequence**: This can expose issues in handling prefix maximums.
   - **Mixed Sequence**: This can expose issues in handling both prefix and suffix maximums and ascents.

3. **Cost Arrays**:
   - **High Values**: Using high values can help expose overflow issues.
   - **Zero Values**: Using zero values can help expose issues in handling zero costs.
   - **Modulo Handling**: Using values close to the modulo can help expose issues in modulo operations.

Given these considerations, let's generate a test input that combines these elements:

```plaintext
3
1 0 1
1 1 1
1 1 1
```

This test input is small-scale and complete. It includes:
- A three-element array, which is a more complex case.
- A cost array with a zero value, which can help expose issues in handling zero costs.
- All other values are set to 1 to simplify the computation and focus on the zero value.

This should help in exposing any issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
MOD = 998244353

def get_factorials(n):
    fact = [1]
    for i in range(1, n + 1):
        fact.append((fact[-1] * i) % MOD)
    return fact

def get_ifactorials(fact):
    ifact = [pow(fact[-1], MOD - 2, MOD)]
    for i in range(len(fact) - 2, -1, -1):
        ifact.append((ifact[-1] * (i + 1)) % MOD)
    return ifact[::-1]

def solve():
    n = int(input())
    a = list(map(int, input().split()))
    b = list(map(int, input().split()))
    c = list(map(int, input().split()))
    fact = get_factorials(n)
    ifact = get_ifactorials(fact)
    a_pref = [0] * n
    a_pref[0] = a[0]
    for i in range(1, n):
        a_pref[i] = (a_pref[i - 1] + a[i]) % MOD
    b_pref = [0] * n
    b_pref[0] = b[0]
    for i in range(1, n):
        b_pref[i] = (b_pref[i - 1] + b[i]) % MOD
    c_pref = [0] * n
    c_pref[0] = c[0]
    for i in range(1, n):
        c_pref[i] = (c_pref[i - 1] + c[i]) % MOD
    dp = [[0] * (n + 1) for _ in range(n + 1)]
    dp[0][0] = c[0]
    for i in range(1, n + 1):
        for j in range(i, -1, -1):
            if j == 0:
                dp[i][j] = (dp[i - 1][j] * a[i - 1] * b[i - 1]) % MOD
            elif j == i:
                dp[i][j] = (dp[i - 1][j - 1] * c[i - 1]) % MOD
            else:
                dp[i][j] = (dp[i - 1][j - 1] * c[i - 1] + dp[i - 1][j] * a[i - 1] * b[i - 1]) % MOD
    ans = [0] * n
    for i in range(n):
        for j in range(i + 1):
            ans[i] += (dp[i][j] * fact[i] * ifact[j] * ifact[i - j]) % MOD
            ans[i] %= MOD
    print(*ans)

if __name__ == "__main__":
    solve()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 0 0 0 1
1 0 0 0 1
1 0 0 0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
MOD = 998244353

def solve(n, a, b, c):
    # Initialize dp array
    dp = [[[0 for _ in range(n)] for _ in range(n+1)] for _ in range(n+1)]

    # Base case
    dp[1][1][0] = 1

    # Compute dp
    for i in range(1, n):
        for p in range(1, i+1):
            for r in range(i):
                if dp[i][p][r] == 0:
                    continue

                # Add a new maximum
                dp[i+1][p+1][r] = (dp[i+1][p+1][r] + dp[i][p][r]) % MOD

                # Add a new minimum
                dp[i+1][p][r+1] = (dp[i+1][p][r+1] + dp[i][p][r]) % MOD

                # Add in between
                dp[i+1][p][r+1] = (dp[i+1][p][r+1] + dp[i][p][r] * (i-1)) % MOD

    # Compute result
    result = [0] * (n+1)
    for i in range(1, n+1):
        for p in range(1, i+1):
            s = i - p + 1  # number of suffix maximums
            for r in range(i):
                result[i] = (result[i] + dp[i][p][r] * a[p-1] * b[s-1] * c[r]) % MOD

    return result[1:]

# Read input
n = int(input())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
c = list(map(int, input().split()))

# Solve and print result
print(*solve(n, a, b, c))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 0 1
1 0 1
1 0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
//#define _GLIBCXX_DEBUG
#include<bits/stdc++.h>

using namespace std;
using E=long long;
constexpr E mod=1004535809;

int n;
vector<E> a,b,c;

template<class A,class B>
void add(A &t,B x){
  t=(t+x)%mod;
}
/*
4
1 2 3 4
1 2 3 4
1 2 3 0
*/

int main(){

  cin>>n;
  a=b=c=vector<E>(n+1);

  for(int i=1; i<=n; i++) cin>>a[i];
  for(int i=1; i<=n; i++) cin>>b[i];
  for(int i=1; i<=n; i++) cin>>c[i-1];

  vector<vector<E>> C(n+1,vector<E>(n+1));
  C[0][0]=1;
  for(int i=1; i<=n; i++){
    for(int j=0; j<=n; j++){
      C[i][j]=C[i-1][j];
      if(j) C[i][j]=(C[i][j]+C[i-1][j-1])%mod;
    }
  }

  vector<vector<vector<E>>> dp(n+1);
  for(int i=0; i<=n; i++){
    dp[i]=vector<vector<E>>(i+1,vector<E>(i+1));
  }
  dp[1][1][0]=1; dp[0][0][0]=1;
  for(int i=1; i<n; i++){
    for(int j=0; j<=i; j++){
      for(int k=0; k<=i; k++){
        if(j<n){
          if(k<n) add(dp[i+1][j+1][k+1],dp[i][j][k]);
        }
        add(dp[i+1][j][k],dp[i][j][k]);
        if(k<n) add(dp[i+1][j][k+1],dp[i][j][k]*1ll*(i-1-k));
        add(dp[i+1][j][k],dp[i][j][k]*1ll*k);
      }
    }
  }

  vector<vector<E>> f(n+1,vector<E>(n+1));
  vector<vector<E>> g(n+1,vector<E>(n+1));

  for(int i=0; i<n; i++){
    for(int j=0; j<i; j++){
      for(int k=0; k<=i; k++){
        f[i][j]=(f[i][j]+dp[i][k][j]*a[k+1])%mod;
      }
    }
  }

  for(int i=0; i<n; i++){
    for(int j=0; j<i; j++){
      for(int k=0; k<=i; k++){
        g[i][j]=(g[i][j]+dp[i][k][j]*b[k+1])%mod;
      }
    }
  }

  vector<vector<E>> coef(n+1,vector<E>(n+1));
  for(int i=1; i<n; i++){
    for(int j=0; j<=n; j++){
      for(int k=1; k+j<=n; k++){
        add(coef[i][j],f[i][k-1]*c[k+j]);
      }
    }
  }

  for(int m=1; m<=n; m++){
    E ans=0;
    if(m==1){
      cout<<a[1]*b[1]%mod*c[0]%mod<<' ';
      continue;
    }
    for(int i=1; i<=m; i++){
      for(int j=1; j<m; j++){
        ans=(ans+dp[m-1][i-1][j-1]*a[i]%mod*b[1]%mod*c[j])%mod;
      }
      for(int j=0; j<m-1; j++){
        ans=(ans+dp[m-1][i-1][m-2-j]*b[i]%mod*a[1]%mod*c[j])%mod;
      }
    }
    for(int t=1; t<=m; t++){
      for(int q=0; q<m-t; q++){
        add(ans,coef[t-1][q]*g[m-t][m-t-1-q]%mod*C[m-1][t-1]%mod);
      }
    }
    cout<<ans<<' ';
  }

  return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
4
1 2 3 4
1 2 3 4
1 2 3 0
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.Scanner;

public class Heartbeat {
    private static final int MOD = 998244353;

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);

        int n = scanner.nextInt();
        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = scanner.nextInt();
        }

        for (int i = 0; i < n; i++) {
            b[i] = scanner.nextInt();
        }

        for (int i = 0; i < n; i++) {
            c[i] = scanner.nextInt();
        }

        scanner.close();

        for (int i = 1; i <= n; i++) {
            int result = calculateCost(Arrays.copyOf(a, i), Arrays.copyOf(b, i), Arrays.copyOf(c, i), i);
            System.out.print(result + " ");
        }
    }

    private static int calculateCost(int[] a, int[] b, int[] c, int n) {
        List<List<Integer>> permutations = generatePermutations(n);
        int totalCost = 0;

        for (List<Integer> perm : permutations) {
            int prefixMaxCount = 0;
            int suffixMaxCount = 0;
            int ascentCount = 0;

            // Calculate prefix maximums
            int currentMax = -1;
            for (int i = 0; i < n; i++) {
                if (perm.get(i) > currentMax) {
                    currentMax = perm.get(i);
                    prefixMaxCount++;
                }
            }

            // Calculate suffix maximums
            currentMax = -1;
            for (int i = n - 1; i >= 0; i--) {
                if (perm.get(i) > currentMax) {
                    currentMax = perm.get(i);
                    suffixMaxCount++;
                }
            }

            // Calculate ascents
            for (int i = 1; i < n; i++) {
                if (perm.get(i) > perm.get(i - 1)) {
                    ascentCount++;
                }
            }

            // Calculate cost for this permutation
            int cost = (int) ((long) a[prefixMaxCount - 1] * b[suffixMaxCount - 1] % MOD * c[ascentCount] % MOD);
            totalCost = (totalCost + cost) % MOD;
        }

        return totalCost;
    }

    private static List<List<Integer>> generatePermutations(int n) {
        List<List<Integer>> result = new ArrayList<>();
        List<Integer> nums = new ArrayList<>();
        for (int i = 1; i <= n; i++) {
            nums.add(i);
        }
        backtrack(result, new ArrayList<>(), nums);
        return result;
    }

    private static void backtrack(List<List<Integer>> result, List<Integer> temp, List<Integer> nums) {
        if (temp.size() == nums.size()) {
            result.add(new ArrayList<>(temp));
        } else {
            for (int i = 0; i < nums.size(); i++) {
                if (temp.contains(nums.get(i))) continue;
                temp.add(nums.get(i));
                backtrack(result, temp, nums);
                temp.remove(temp.size() - 1);
            }
        }
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 0 1 0 1
0 1 0 1 0
1 1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
// LUOGU_RID: 171345444
  #include<bits/stdc++.h>
  using namespace std;
  const int N=705,mod=998244353;
  int n;
  int a[N],b[N],c[N];
  long long f[2][N][N];
  long long u[N][N],v[N][N],w[N][N];
  long long C[N][N];
  signed main(){
      scanf("%d",&n);
      for(int i=0;i<=n;i++){
          C[i][0]=1;
          for(int j=1;j<=i;j++){
              C[i][j]=(C[i-1][j-1]+C[i-1][j])%mod;
          }
      }
      for(int i=1;i<=n;i++){
          scanf("%d",&a[i]);
      }
      for(int i=1;i<=n;i++){
          scanf("%d",&b[i]);
      }
      for(int i=0;i<n;i++){
          scanf("%d",&c[i]);
      }
      u[0][0]=a[1],v[0][0]=b[1];
      for(int i=1;i<=n;i++){
          if(i==1){
              f[1][1][0]=1;
          }
          else{
              for(int j=1;j<=i;j++){
                  for(int k=0;k<i;k++){
                      f[i&1][j][k]=0;
                  }
              }
              for(int j=1;j<i;j++){
                  for(int k=0;k<i-1;k++){
                      f[i&1][j][k]=(f[i&1][j][k]+f[(i-1)&1][j][k]*(k+1)%mod)%mod;
                      f[i&1][j][k+1]=(f[i&1][j][k+1]+f[(i-1)&1][j][k]*(i-1-k-1)%mod)%mod;
                      f[i&1][j+1][k+1]=(f[i&1][j+1][k+1]+f[(i-1)&1][j][k])%mod;
                  }
              }
          }
          for(int j=1;j<=i;j++){
              for(int k=0;k<i;k++){
                  u[i][k]=(u[i][k]+f[i&1][j][k]*a[j+1]%mod)%mod;
                  v[i][k]=(v[i][k]+f[i&1][j][i-1-k]*b[j+1]%mod)%mod;
              }
          }
      }
      for(int p=0;p<=n;p++){
          for(int x=0;x<=p;x++){
              for(int y=0;x+y<n;y++){
                  w[p][y]=(w[p][y]+u[p][x]*c[x+y+(p>0)]%mod)%mod;
              }
          }
      }
      for(int i=1;i<=n;i++){
          long long ans=0;
          for(int p=1;p<=i;p++){
              for(int y=0;y<=i-p;y++){
                  ans=(ans+w[p-1][y]*v[i-p][y]%mod*C[i-1][p-1]%mod)%mod;
              }
          }
          printf("%lld ",ans);
      }
      return 0;
  }

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 0 0 0 1
1 0 0 0 1
1 0 0 0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
#include<bits/stdc++.h>
#define ll long long
#define mod 998244353
#define N 705
using namespace std;
ll n,i,j,k,l,las,cur,a[N],b[N],c[N],f[2][N][N],u[N][N],v[N][N],C[N][N],vv[N][N],ans[N],jc[N],inv[N];
inline void add(ll &a,ll b){(a+=b)>=mod&&(a-=mod);}
inline ll qmi(ll a,ll b,ll p){
	ll res = 1%p,t = a;
	while(b){
		if(b&1) res=res*t%p;
		t=t*t%p;
		b>>=1;
	}
	return res;
}
int main(){
	ios::sync_with_stdio(false);
	cin>>n;
	jc[0]=1;
	for(i=1;i<=n;i++) jc[i]=jc[i-1]*i%mod;
	inv[n]=qmi(jc[n],mod-2,mod);
	for(i=n;i>=1;i--) inv[i-1]=inv[i]*i%mod;
	C[0][0]=1;
	for(i=0;i<=n;i++){
		for(j=0;j<=n;j++){
			if(i) C[i][j]+=C[i-1][j];
			if(i&&j) C[i][j]+=C[i-1][j-1];
			if(C[i][j]>=mod) C[i][j]-=mod;
		}
	}
	for(i=1;i<=n;i++) cin>>a[i];
	for(i=1;i<=n;i++) cin>>b[i];
	for(i=0;i<n;i++) cin>>c[i];
	f[1][1][0] = 1,u[0][0] = a[1],v[0][0] = b[1];
	for(i=1;i<=n;i++){
		for(j=0;j<=n;j++){
			for(k=0;k<i;k++){
				u[i][k]=(u[i][k]+f[i&1][j][k]*a[j+1])%mod;
				v[i][i-1-k]=(v[i][i-1-k]+f[i&1][j][k]*b[j+1])%mod;
			}
		}
		las=(i&1),cur=((i+1)&1);
		for(j=0;j<=n;j++) for(k=0;k<=n;k++) f[cur][j][k]=0;
		for(j=0;j<=n;j++){
			for(k=0;k<=n;k++){
				if(!f[las][j][k]) continue;
				add(f[cur][j+1][k+1],f[las][j][k]);
				add(f[cur][j][k],f[las][j][k]);
				add(f[cur][j][k+1],f[las][j][k]*(i-1-k)%mod);
				add(f[cur][j][k],f[las][j][k]*k%mod);
			}
		}
	}
	for(i=0;i<=n;i++) for(j=0;j<=n;j++) for(k=0;k<=n;k++) add(vv[i][k],u[i][j]*c[j+k+(i>0)]%mod*inv[i]%mod);
	for(i=0;i<=n;i++) for(j=0;j<=n;j++) for(k=0;k<=n;k++) add(ans[i+j],v[j][k]*vv[i][k]%mod*inv[j]%mod);
//	for(i=0;i<=n;i++) for(j=0;j<=n;j++) for(k=0;k<=n;k++) for(l=0;l<=n;l++) add(ans[i+j],u[i][k]*v[j][l]%mod*C[i+j][i]%mod*c[k+l+(i>0)]%mod);
	for(i=1;i<=n;i++) cout<<ans[i-1]*jc[i-1]%mod<<" ";
	cout<<endl;
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 0 1
1 0 1
1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
from itertools import permutations

MOD = 998244353

def compute_f(n, a, b, c):
    results = []

    for size in range(1, n + 1):
        total_cost = 0
        perms = permutations(range(1, size + 1))

        for perm in perms:
            prefix_max_count = 0
            suffix_max_count = 0
            ascent_count = 0

            # Count prefix maximums
            current_max = -float('inf')
            for i in range(size):
                if perm[i] > current_max:
                    prefix_max_count += 1
                    current_max = perm[i]

            # Count suffix maximums
            current_max = -float('inf')
            for i in range(size - 1, -1, -1):
                if perm[i] > current_max:
                    suffix_max_count += 1
                    current_max = perm[i]

            # Count ascents
            for i in range(1, size):
                if perm[i] > perm[i - 1]:
                    ascent_count += 1

            # Ensure indices are within bounds
            if prefix_max_count - 1 < len(a) and suffix_max_count - 1 < len(b) and ascent_count < len(c):
                cost = a[prefix_max_count - 1] * b[suffix_max_count - 1] * c[ascent_count]
                total_cost = (total_cost + cost) % MOD

        results.append(total_cost)

    return results

# Read input
n = int(input().strip())
a = list(map(int, input().strip().split()))
b = list(map(int, input().strip().split()))
c = list(map(int, input().strip().split()))

# Ensure the lengths of a, b, c are at least n
if len(a) < n or len(b) < n or len(c) < n:
    raise ValueError("Input arrays a, b, or c are too short")

results = compute_f(n, a, b, c)
for result in results:
    print(result, end=' ')

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the given code, we need to focus on edge cases and scenarios that can maximize the failure probability. Here are some key points to consider:

1. **Small Input Size**: Start with a small input size to ensure the test input is complete and valid.
2. **Edge Cases**:
   - **Single Element Array**: Test with `n = 1` to ensure the code handles the base case correctly.
   - **Two Elements Array**: Test with `n = 2` to ensure the code handles the simplest non-trivial case.
   - **Three Elements Array**: Test with `n = 3` to ensure the code handles a slightly more complex case.
3. **Special Values**:
   - **All Zeros**: Test with all zero values in the cost arrays to ensure the code handles zero costs correctly.
   - **Large Values**: Test with large values in the cost arrays to ensure the code handles large numbers and modulo operations correctly.

Let's generate a test input that includes these edge cases and special values:

```plaintext
3
0 0 0
0 0 0
0 0
```

This test input will help expose any issues with the code, especially in handling zero values and small input sizes. Here's the step-by-step reasoning:

- **n = 3**: This is a small input size that is complete and valid.
- **a = [0, 0, 0]**: All elements are zero, which should result in a total cost of zero for all permutations.
- **b = [0, 0, 0]**: Similarly, all elements are zero.
- **c = [0, 0]**: All elements are zero, and the length is `n-1` as required.

This input should result in all `f(i)` values being zero, which can help identify any issues in the code's handling of zero values and small input sizes.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Heartbeat
time_limit_ms: 5000
memory_limit_mb: 512
Description: For an array $u_1, u_2, \ldots, u_n$, define

  * a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$;    * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$;    * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Input: The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Output: Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

Examples:
input:
3
1 1 1
1 1 1
1 1 1
output:
1 2 6

input:
3
1 2 3
2 3 1
3 1 2
output:
6 13 34

input:
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
output:
600000000 303511294 612289529 324650937 947905622

Note: In the second example:

  * Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$.    * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$.    * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$.    * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$.    * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

Buggy Code:
import itertools

mod = 998244353

def hisobala1(p):
    count = 0
    mv = -1
    for v in p:
        if v > mv:
            count += 1
            mv = v
    return count

def hisobala2(p):
    s = 0
    mv = -1
    for v in reversed(p):
        if v > mv:
            s += 1
            mv = v
    return s

def hisobala3(p):
    s = 0
    for i in range(1, len(p)):
        if p[i] > p[i-1]:
            s += 1
    return s

def solve(n, a, b, c):
    c1 = []
    for kk in range(1, n+1):
        pcs = 0
        for p in itertools.permutations(range(1, kk+1)):
            pm = hisobala1(p)
            sm = hisobala2(p)
            a1 = hisobala3(p)
            pc = a[pm-1] * b[sm-1] % mod * c[a1] % mod
            pcs = (pcs + pc) % mod
        c1.append(pcs)
    return c1

n = int(input())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
c = list(map(int, input().split()))

res = solve(n, a, b, c)
print(' '.join(map(str, res)))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
4
1 2 3 4
4 3 2 1
1 2 3 4
```