**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import sys
from collections import defaultdict,deque
import bisect
import  math
from copy import deepcopy
input=sys.stdin.readline
# for _ in range(int(input())):
m,n=map(int,input().strip().split())
l1=list(map(int,input().strip().split()))
l2=list(map(int,input().strip().split()))
short=sum(l1)+sum(l2);long=sum(l1)
s1,s2=0,0
for i in range(m):
    if l1[i]!=0 and l2[i]!=0:
        s1+=2;s2+=2
    elif l1[i]!=0 :
        s1+=1
    elif l2[i]!=0 :
        s2+=1
# print(short,long)
# print(s1,s2)
if n==1:
    print((l1[0]*short+l2[0]*long)%1000000007)
else:
    c1=1;c2=1
    while n>1:
        c1*=s1;c2*=s2
        n-=1
    # print(c1,c2)

    print((short*c1+long*c2)%1000000007)

**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 3
1 0 1
0 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import sys
from collections import defaultdict,deque
import bisect
import  math
from copy import deepcopy
input=sys.stdin.readline
# for _ in range(int(input())):
m,n=map(int,input().strip().split())
l1=list(map(int,input().strip().split()))
l2=list(map(int,input().strip().split()))
short=sum(l1)+sum(l2);long=sum(l1)
s1,s2=0,0
for i in range(m):
    if l1[i]!=0 and l2[i]!=0:
        s1+=2;s2+=2
    elif l1[i]!=0 :
        s1+=1
    elif l2[i]!=0 :
        s2+=1
# print(short,long)
# print(s1,s2)
if n==1:
    print(l1[0]*short+l2[0]*long)
else:
    c1=1;c2=1
    while n>1:
        c1*=s1;c2*=s2
        n-=1
    # print(c1,c2)

    print(short*c1+long*c2)

**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 3
1 0 0
0 1 0
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
m, n = map(int, input().split())
s = [0]+list(map(int, input().split()))
l = [0]+list(map(int, input().split()))

S = sum(s)
L = sum(l)

dp = [[0]*(m+1) for _ in range(n+1)]

for j in range(1, m+1):
    dp[n][j] = s[j]*(S+L)+l[j]*S

for i in range(n-1, 0, -1):
    for j in range(1, m+1):
        for k in range(1, m+1):
            dp[i][j] += (s[j]*(s[k]+l[k])+l[j]*s[k])*dp[i+1][k]

print(dp[1][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
2 1000
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: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.util.*;

public class E_1_Trails_Easy {
    public static int[] s;
    public static int[] l;
    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int m = in.nextInt();
        int n = in.nextInt();
         s = new int[m];
         l =  new int[m];
        for (int i = 0 ;i<s.length;i++)s[i]=in.nextInt();
        for (int i =0 ;i<l.length;i++)l[i] = in.nextInt();
        long[][] dp = new long[n][m];
        //for (long[] ele : dp)System.out.println(Arrays.toString(l));
        for (long[] ele : dp)Arrays.fill(ele,-1);
        System.out.println( memo(dp,0,0));
    }
    public static int M = (int)1e9+7;
    public static long memo(long[][] dp , int i, int m){
        //System.out.println(i+" "+m);
        if (i==dp.length)return 1;
        //if (dp[i][m]!=-1)return dp[i][m];
        long smallpath = 0;
        long largepath = 0;
        for (int j = 0 ;j<dp[0].length;j++){
            smallpath += s[m]*(s[j]+l[j])%M*memo(dp, i+1, j)%M;
            //System.out.println(smallpath);
            smallpath%=M;
            largepath += l[m]*(s[j])%M*memo(dp, i+1, j)%M;
            largepath%=M;
        }
        //System.out.println(smallpath);
        return dp[i][m]=(smallpath+largepath)%M;
    }

}

**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
2 3
1 0
0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.*;

public class trails {

    static int mod = (int)1e9+7;
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter pw = new PrintWriter(System.out);

        StringTokenizer st = new StringTokenizer(br.readLine());
        int m = Integer.parseInt(st.nextToken());
        int n = Integer.parseInt(st.nextToken());

        int[][] arr = new int[m][2];

        st = new StringTokenizer(br.readLine());
        for (int i = 0; i < m; i++) {
            arr[i][0] = Integer.parseInt(st.nextToken());
        }

        st = new StringTokenizer(br.readLine());
        for (int i = 0; i < m; i++) {
            arr[i][1] = Integer.parseInt(st.nextToken());
        }

        long[][] dp = new long[m][n+1];
        for(int i=0;i<dp.length;i++){
            Arrays.fill(dp[i],-1);
        }

        long ans = helper(0,n,dp,arr);
        ans%=mod;
        System.out.println(ans);
    }
    public static long helper(int m , int n , long[][] dp , int[][] arr){
        if(n==0) return 1;

        if(dp[m][n]!=-1) return dp[m][n];

        long ans=0;

        for(int i=0;i<arr.length;i++){
            long temp=0;
            temp+= ((long)arr[m][0] * arr[i][0] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            temp+= ((long)arr[m][0] * arr[i][1] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            temp+= ((long)arr[m][1] * arr[i][0] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            ans += (temp%mod);
            ans%=mod;
        }

        return dp[n][m]=ans;
    }
}

**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 that can maximize the probability of failure. Here are some considerations:

1. **Boundary Conditions**: Test with the minimum and maximum values for `m` and `n`.
2. **Zero and Maximum Values**: Test with `s_i` and `l_i` being 0 or at their maximum value (1000).
3. **Symmetry and Asymmetry**: Test with symmetric and asymmetric distributions of short and long trails.

Given the constraints:
- \(1 \le m \le 10^2\)
- \(1 \le n \le 10^3\)
- \(0 \le s_i, l_i \le 10^3\)

A small-scale test input that can be challenging is:

- Set \(m = 2\) and \(n = 2\).
- Set \(s_1 = 1\) and \(s_2 = 0\) (one cabin with a short trail, one cabin with no short trails).
- Set \(l_1 = 0\) and \(l_2 = 1\) (one cabin with no long trails, one cabin with a long trail).

This setup ensures that the algorithm must handle a cabin with no trails and a cabin with only one type of trail, which can expose issues in the recursive logic and the dynamic programming table.

Here is the test input:

```plaintext
2 2
1 0
0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include<bits/stdc++.h>
#define int long long
using namespace std;

int rec(int ind,int days,int n,vector<int>&v1,vector<int>&v2){
  int ans=0;
  // cout<<ind<<" "<<days<<endl;
  int m=v1.size();
  if(days==0)return 1;
  // cout<<ind<<" "<<days<<endl;
  if(v1[ind]>0){
    for(int i=0;i<m;i++){
        ans=(ans+v1[ind]*(v1[i]+v2[i])*rec(i,days-1,n,v1,v2));
    }
  }
  for(int i=0;i<m;i++){
    ans=(ans+v2[ind]*v1[i]*rec(i,days-1,n,v1,v2));
  }
  return ans;
}

void solve(){
    int m,n;cin>>m>>n;
    vector<int>v1(m),v2(m);
    for(int i=0;i<m;i++){
        cin>>v1[i];
    }
    for(int i=0;i<m;i++){
        cin>>v2[i];
    }
   cout<<rec(0,n,n,v1,v2)<<endl;
}

signed main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int t;
    // cin>>t;
    t=1;
    while(t--){
       solve();
    }
    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 expose the bug in the provided code, we need to consider edge cases that might cause the recursive function to fail. Here are some key points to consider:

1. **Zero Trails**: If a cabin has zero short and zero long trails, the recursive function might not handle this correctly.
2. **Large Number of Days**: The code might have issues with large values of `n` due to the recursive nature and potential overflow.
3. **Single Cabin**: If there is only one cabin, the recursion should handle this case correctly.
4. **All Short Trails**: If all cabins have only short trails, the recursion should handle this case efficiently.
5. **All Long Trails**: If all cabins have only long trails, the recursion should handle this case and return 0 for any `n > 1`.

Given these considerations, let's create a test input that combines some of these edge cases to maximize the probability of exposing the bug.

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

### Explanation:
- **3 Cabins**: This is a small-scale input to ensure it fits within the constraints.
- **3 Days**: This is a moderate number of days to ensure the recursion goes deep enough to potentially expose bugs.
- **Trails**:
  - Cabin 1 has 0 short trails and 1 long trail.
  - Cabin 2 has 1 short trail and 0 long trails.
  - Cabin 3 has 0 short trails and 1 long trail.

This setup ensures that:
- Cabin 1 can only take a long trail to the lake.
- Cabin 2 can only take a short trail to the lake.
- Cabin 3 can only take a long trail to the lake.

This configuration will test how the code handles a mix of short and long trails, especially when one cabin has only long trails. The recursive function should correctly handle the constraints and return the correct number of combinations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using pii = pair<ll, ll>;
using i128 = __int128_t;
#define fi first
#define se second
using ll = long long;
using ull = unsigned long long;
using i64 = long long;
std::mt19937_64 rng{std::chrono::steady_clock::now().time_since_epoch().count()};
const int mod = 1e9 + 7;
void solve()
{
    int m, n;
    cin >> m >> n;
    vector<ll> s(m + 1), l(m + 1);
    vector<ll> ps(m + 1), pl(m + 1);
    for (int i = 1; i <= m; i++)
    {
        cin >> s[i];
        ps[i] = s[i] + ps[i - 1];
    }
    for (int i = 1; i <= m; i++)
    {
        cin >> l[i];
        pl[i] = l[i] + pl[i - 1];
    }
    vector dp(n + 1, vector<ll>(m + 1));
    dp[0][1] = 1;
    for (int i = 1; i <= n; i++)
    {
        for (int j = 1; j <= m; j++)
        {
            for (int k = 1; k <= m; k++)
            {
                dp[i][j] = (dp[i][j] + (dp[i - 1][k] * ((s[k] ? s[j] + l[j] : 0) + (l[k] ? s[j] : 0)))) % mod;
            }
        }
    }
    ll ans = 0;
    for (int i = 1; i <= m; i++)
    {
        ans = (ans + dp[n][i]) % mod;
    }
    cout << ans << '\n';
}
int main()
{
    ios::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
    int t = 1;
    // cin >> t;
    while (t--)
    {
        solve();
    }
    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 code, we need to focus on edge cases and scenarios that can maximize the failure probability. Here are the steps to consider:

1. **Edge Cases**:
   - **Single Cabin**: Test with a single cabin to see if the algorithm handles it correctly.
   - **No Short Trails**: Test with a cabin that has no short trails to ensure the algorithm handles it correctly.
   - **No Long Trails**: Test with a cabin that has no long trails to ensure the algorithm handles it correctly.
   - **Maximum Number of Trails**: Test with the maximum number of trails (1000) to ensure the algorithm handles large numbers correctly.

2. **Complex Scenarios**:
   - **Multiple Cabins with Mixed Trails**: Test with multiple cabins, each having a mix of short and long trails.
   - **High Number of Days**: Test with a high number of days to ensure the algorithm handles large values of \( n \) correctly.

Given the constraints, we will generate a small-scale test input that is complete and valid:

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

This test input is designed to:
- Test a scenario with 3 cabins.
- Test a cabin with no short trails (cabin 2).
- Test a cabin with no long trails (cabin 1).
- Test a cabin with both short and long trails (cabin 3).
- Test with a moderate number of days (3) to ensure the algorithm handles multiple days correctly.

This input should help expose any issues in the buggy code, particularly in handling cabins with no short or long trails and in the dynamic programming logic.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
MOD = 10**9 + 7

def mat_mult(A, B, size):
    # Matrix multiplication (A * B) % MOD
    result = [[0] * size for _ in range(size)]
    for i in range(size):
        for j in range(size):
            result[i][j] = sum(A[i][k] * B[k][j] for k in range(size)) % MOD
    return result

def mat_pow(matrix, power, size):
    # Matrix exponentiation (matrix ^ power) % MOD
    result = [[1 if i == j else 0 for j in range(size)] for i in range(size)]
    base = matrix

    while power:
        if power % 2 == 1:
            result = mat_mult(result, base, size)
        base = mat_mult(base, base, size)
        power //= 2

    return result

def hiking_trails(m, n, s, l):
    t = [s[i] + l[i] for i in range(m)]

    # Create the transition matrix A
    A = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            A[i][j] = (t[i] * t[j] - l[i] * l[j]) % MOD

    # Exponentiate the transition matrix
    A_n = mat_pow(A, n, m)

    # Initial state vector (Harry starts at cabin 1)
    initial_state = [0] * m
    initial_state[0] = 1

    # Compute the final state after n days
    final_state = [0] * m
    for i in range(m):
        final_state[i] = sum(A_n[i][j] * initial_state[j] for j in range(m)) % MOD

    # The result is the sum of the final state
    result = sum(final_state) % MOD
    return result

gg = list(map(int, input().split()))
m = gg[0]
n = gg[1]
s =  list(map(int, input().split()))
l =  list(map(int, input().split()))
# s = [1, 0, 1]
# l = [0, 1, 1]
print(hiking_trails(m, n, s, l))

**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
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: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
MOD = 10**9 + 7

def matrix_mult(A, B, m):
    return [[sum(A[i][k] * B[k][j] for k in range(m)) % MOD for j in range(m)] for i in range(m)]

def matrix_expo(A, n, m):
    res = [[1 if i == j else 0 for j in range(m)] for i in range(m)]
    while n:
        if n % 2:
            res = matrix_mult(res, A, m)
        A = matrix_mult(A, A, m)
        n //= 2
    return res

def count_possible_combinations(m, n, s, l):
    # Construct the transition matrix
    T = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            T[i][j] = s[i] * (s[j] + l[j]) + l[i] * s[j]
            T[i][j] %= MOD

    # Use matrix exponentiation to find T^n
    T_n = matrix_expo(T, n, m)

    # Initial state vector (only starting from cabin 1 is 1)
    initial_state = [1] + [0] * (m - 1)

    # Calculate the final state after n days
    final_state = [sum(T_n[i][j] * initial_state[j] for j in range(m)) % MOD for i in range(m)]

    return sum(final_state) % MOD

# Read input from the user
def main():
    m, n = map(int, input().split())
    s = list(map(int, input().split()))
    l = list(map(int, input().split()))

    # Calculate the result
    result = count_possible_combinations(m, n, s, l)
    print(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:
```plaintext
2 1
1 0
0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.util.*;
import java.io.*;

public class cf2 {
	static final int MOD = 1000000000+7;
	static class FastScanner {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		StringTokenizer st = new StringTokenizer("");

		String next() {
			while (!st.hasMoreTokens())
				try {
					st = new StringTokenizer(br.readLine());
				} catch (IOException e) {
					e.printStackTrace();
				}
			return st.nextToken();
		}

		int nextInt() {
			return Integer.parseInt(next());
		}

		int[] readArray(int n) {
			int[] a = new int[n];
			for (int i = 0; i < n; i++) a[i] = nextInt();
			return a;
		}

		long nextLong() {
			return Long.parseLong(next());
		}
	}

	public static void main(String[] args) throws IOException {
		FastScanner fs = new FastScanner();
		PrintWriter pw = new PrintWriter(new OutputStreamWriter(System.out));
		int m = fs.nextInt(), n = fs.nextInt();
		int[] s = fs.readArray(m);
		int[] l = fs.readArray(m);
		long[][] dp = new long[n+1][m];
		dp[0][0] = 1;
		for(int i = 1; i <= n; i++){
			for(int j = 0; j < m; j++){
				for(int k = 0; k < m; k++){
					long num = (((((dp[i-1][k]*s[k])%MOD)*l[j])%MOD +
							(((dp[i-1][k]*l[k])%MOD)*s[j])%MOD)%MOD +
							(((dp[i-1][k]*s[k])%MOD)*s[j])%MOD)%MOD;
					dp[i][j] = (dp[i][j] + num)%MOD;
				}
			}
		}
		long answer = 0;
		for(long comb : dp[n]) answer = (answer+comb)%MOD;
		pw.println(answer);
		pw.close();
	}
}

**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 can highlight potential issues. Here are the steps to think through this:

1. **Edge Cases for \( m \) and \( n \):**
   - \( m \) and \( n \) should be within the given constraints but should be small enough to be manageable.
   - Consider \( m = 2 \) and \( n = 3 \) to keep the input small but non-trivial.

2. **Edge Cases for \( s_i \) and \( l_i \):**
   - Consider scenarios where \( s_i \) and \( l_i \) are at the boundaries of the constraints.
   - Consider cases where some \( s_i \) and \( l_i \) are zero to test the handling of such cases.
   - Consider cases where all \( s_i \) and \( l_i \) are non-zero to test the general algorithm.

3. **Combination of Trails:**
   - Ensure that the combination of short and long trails between cabins and the lake can create a complex enough scenario to expose potential bugs.

Given these considerations, let's create a test input:

```plaintext
2 3
1 0
0 1
```

This input:
- \( m = 2 \) (two cabins)
- \( n = 3 \) (three days)
- \( s_1 = 1 \), \( s_2 = 0 \) (Cabin 1 has one short trail, Cabin 2 has no short trails)
- \( l_1 = 0 \), \( l_2 = 1 \) (Cabin 1 has no long trails, Cabin 2 has one long trail)

This input is small but should expose issues in the algorithm, particularly in handling cases where one cabin has only short trails and the other has only long trails. The dynamic programming approach in the code should correctly handle these edge cases, but if there are any bugs, this input will likely reveal them.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.util.*;

public class trails {

    static int mod = (int)1e9+7;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);

        int m =sc.nextInt();
        int n=sc.nextInt();

        int[][] arr = new int[m][2];

        for(int i=0;i<m;i++){
            arr[i][0]=sc.nextInt();
        }
        for(int i=0;i<m;i++){
            arr[i][1]=sc.nextInt();
        }

        long[][] dp = new long[m][n+1];
        for(int i=0;i<dp.length;i++){
            Arrays.fill(dp[i],-1);
        }

        long ans = helper(0,n,dp,arr);
        ans%=mod;
        System.out.println(ans);
    }
    public static long helper(int m , int n , long[][] dp , int[][] arr){
        if(n==0) return 1;

        if(dp[m][n]!=-1) return dp[m][n];

        long ans=0;

        for(int i=0;i<arr.length;i++){
            long temp=0;
            temp+= ((long)arr[m][0] * arr[i][0] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            temp+= ((long)arr[m][0] * arr[i][1] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            temp+= ((long)arr[m][1] * arr[i][0] * helper(i,n-1,dp,arr))%mod;
            temp%=mod;

            ans += (temp%mod);
            ans%=mod;
        }

        return dp[n][m]=ans;
    }
}

**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 3
1 0 1
0 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include <bits/stdc++.h>
#define int long long
using namespace std;
const int M=1e2+7;
const int N=1e3+7;
const int MD=1e9+7;
int m,n,s[M],l[M],dp[N][M],cnt;
int32_t main(){
	cin.tie(NULL)->sync_with_stdio(false);
	cin>>m>>n;
	for(int i=1; i<=m; i++){
		cin>>s[i];
	}
	for(int i=1; i<=m; i++){
		cin>>l[i];
		if(i==1) dp[0][i]=1;
		else dp[0][i]=0;
	}
	for(int i=1; i<=n; i++){
		for(int j=1; j<=m; j++){
			for(int k=1; k<=m; k++){
				dp[i][j]+=(dp[i-1][k]*((s[k]*(s[j]+l[j]))%MD+(l[k]*s[j])%MD))%MD;
				dp[i][j]%=MD;
			}
		}
	}
	for(int i=1; i<=m ;i++){
		cnt+=dp[n][i];
	}
	cout<<cnt;
}

**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 provided code, we need to consider edge cases and scenarios that can maximize the probability of failure. The bug in the code might be related to how it handles the constraints, especially when the number of short and long trails are zero or when the number of days is large.

Let's consider the following scenarios:
1. **No Short Trails from Cabin 1**: This could cause issues because the code might not handle the case where Harry cannot start his hike from Cabin 1 if there are no short trails.
2. **No Long Trails from Any Cabin**: This could simplify the problem but might still expose issues in the dynamic programming (DP) table initialization.
3. **Large Number of Days**: This can test the efficiency and correctness of the DP transitions.

Let's create a small-scale test input that fits these criteria:

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

Explanation:
- **m = 3**: There are 3 cabins.
- **n = 3**: Harry walks for 3 days.
- **s = [0, 1, 1]**: Cabin 1 has 0 short trails, Cabin 2 has 1 short trail, and Cabin 3 has 1 short trail.
- **l = [1, 0, 1]**: Cabin 1 has 1 long trail, Cabin 2 has 0 long trails, and Cabin 3 has 1 long trail.

This input:
- Ensures that Cabin 1 has no short trails, which means Harry must start his hike from a cabin that has a short trail.
- Ensures that Cabin 2 and Cabin 3 have short trails, allowing Harry to move to these cabins.
- Ensures that Cabin 1 has a long trail, which can be used to return to the lake.
- Tests the code over 3 days to ensure the DP transitions are correct.

This test input should help expose any issues in the code related to handling the absence of short trails from the starting cabin and the transitions over multiple days.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

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

#define all(v) v.begin(), v.end()
#define rall(v) v.rbegin(), v.rend()
#define size(v) (int)v.size()

using namespace std;
using ll = long long;

const int MOD = 1e9 + 7;

int main() {
    cin.tie(0), ios::sync_with_stdio(0);

    int M, N;
    cin >> M >> N;

    vector<int> S(M), L(M);
    for (int& e : S) cin >> e;
    for (int& e : L) cin >> e;

    vector dp(N + 1, vector<ll>(M));
    dp[0][0] = 1;

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < M; j++) {
            for (int k = 0; k < M; k++) {
                dp[i + 1][k] += dp[i][j] * (S[j] + L[j]) % MOD * S[k] % MOD;
                dp[i + 1][k] %= MOD;
                dp[i + 1][k] += dp[i][j] * S[j] % MOD * L[k] % MOD;
                dp[i + 1][k] %= MOD;
            }
        }
    }

    cout << accumulate(all(dp[N]), 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
2 1000
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: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
from functools import lru_cache

m, n = [int(c) for c in input().strip().split()]
s = [int(c) for c in input().strip().split()]
l = [int(c) for c in input().strip().split()]

@lru_cache(maxsize=None)
def trail(i, j):
    return 1 if not j else sum([(s[i] * (l[k] + s[k]) + (l[i] * s[k])) * trail(k, j-1) for k in range(m)])

print(trail(0, n))

**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 issues in the algorithm. Here are the steps to consider:

1. **Edge Cases for \( m \) and \( n \)**:
   - \( m \) and \( n \) should be at the lower and upper bounds to test the limits of the algorithm.
   - \( m = 1 \) and \( n = 1 \) to test the simplest case.
   - \( m = 100 \) and \( n = 1000 \) to test the upper bounds.

2. **Edge Cases for \( s_i \) and \( l_i \)**:
   - \( s_i \) and \( l_i \) should be at the lower and upper bounds to test the limits of the algorithm.
   - \( s_i = 0 \) and \( l_i = 0 \) to test the case where no trails are available.
   - \( s_i = 1000 \) and \( l_i = 1000 \) to test the case with the maximum number of trails.

3. **Scenarios to Expose Bugs**:
   - Test cases where \( s_i \) and \( l_i \) are mixed to ensure the algorithm handles combinations correctly.
   - Test cases where the starting cabin has no short trails to ensure the algorithm handles this constraint correctly.

Given these considerations, a small-scale test input that is complete and valid and can expose potential bugs is:

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

This test input:
- \( m = 3 \) and \( n = 3 \) to ensure the algorithm handles a moderate number of days and cabins.
- \( s_1 = 0 \) and \( l_1 = 1 \) to test the case where the starting cabin has no short trails.
- \( s_2 = 1 \) and \( l_2 = 0 \) to test the case where a cabin has only short trails.
- \( s_3 = 1 \) and \( l_3 = 1 \) to test the case where a cabin has both short and long trails.

This input should help expose any issues in the algorithm's handling of the constraints and combinations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import sys
input = sys.stdin.readline

mod = 10 ** 9 + 7

m, n = map(int, input().split())
short = list(map(int, input().split()))
long  = list(map(int, input().split()))

def MMul(A, B):
    p, q, r = len(A), len(B), len(B[0])
    res = [[0 for _ in range(r)] for _ in range(p)]
    for i in range(m):
        for j in range(r):
            for k in range(q):
                res[i][j] += (A[i][k] * B[k][j]) % mod
                res[i][j] %= mod

    return res

def MPow(A, n):
    p = len(A)
    tmp = A
    res = [[0] * p for _ in range(p)]
    for i in range(p):
        res[i][i] = 1

    i = 0
    while (1 << i) <= n:
        if (1 << i) & n:
            res = MMul(res, tmp)
        tmp = MMul(tmp, tmp)
        i += 1

    return res

A = [[0 for _ in range(m)] for _ in range(m)]

for i in range(m):
    for j in range(m):
        A[i][j] = ((short[i] + long[i]) * (short[j] + long[j]) - long[i] * long[j]) % mod

x = [[0] for _ in range(m)]
x[0][0] = 1

res = MMul(MPow(A, n), x)
print(sum(sum(l) for l in 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:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider the following:

1. **Edge Cases:**
   - **Single Cabin:** This can help check if the code handles the simplest case correctly.
   - **No Short Trails:** This can help check if the code handles the case where no short trails are available.
   - **No Long Trails:** This can help check if the code handles the case where no long trails are available.
   - **Maximum Trails:** This can help check if the code handles large values correctly.

2. **Complex Scenarios:**
   - **Multiple Cabins with Varying Trails:** This can help check if the code handles multiple cabins with different combinations of short and long trails.
   - **Large Number of Days:** This can help check if the code handles a large number of days correctly.

Given the constraints, a small-scale test input that is complete and valid should be generated. Here, we will focus on a scenario with multiple cabins and a large number of days to maximize the failure probability in faulty solutions.

```plaintext
5 10
1 2 0 3 1
0 1 3 0 2
```

Explanation:
- **m = 5:** There are 5 cabins.
- **n = 10:** Harry walks for 10 days.
- **Short Trails:** The number of short trails for each cabin is [1, 2, 0, 3, 1].
- **Long Trails:** The number of long trails for each cabin is [0, 1, 3, 0, 2].

This test input includes:
- A cabin with no short trails (cabin 3).
- A cabin with no long trails (cabin 1 and cabin 4).
- Multiple cabins with varying numbers of short and long trails.
- A large number of days to ensure the matrix exponentiation and multiplication operations are thoroughly tested.

This should help expose any bugs in the matrix operations and the handling of different trail configurations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.util.*;
import java.util.function.BinaryOperator;
import java.lang.*;
import java.lang.reflect.Array;
import java.io.*;
public class Main
{
    static final PrintWriter out =new PrintWriter(System.out);
    static final FastReader sc = new FastReader();
     static class Pair{
          int f; int s;
          public Pair(int first,int second)
          {
                this.f=first;
                this.s=second;
          }
    }
    static class Triple{
          int f,s,t;
          public Triple(int first,int second,int third)
          {
                this.f=first;
                this.s=second;
                this.t=third;
          }
    }
    public static boolean prime(long n)
    {
        if(n==1L) return false;
        if(n==2L) return true;
        for(long i=2;i*i<=n;i++)
        if(n%i==0) return false;
        return true;
    }
    public static long mod=(long)1e9+7;
    public static ArrayList<Integer> ans;
	public static void main (String[] args) throws java.lang.Exception
	{
        int tes=1;
        StringBuilder sb=new StringBuilder();
	    lable:for(int fuck=1;fuck<=tes;fuck++)
	    {
            int n=sc.nextInt(),m=sc.nextInt(),i;
            long s[]=new long[n];
            long l[]=new long[n];
            for(i=0;i<n;i++) s[i]=sc.nextLong();
            for(i=0;i<n;i++) l[i]=sc.nextLong();
            Long dp[][][]=new Long[n][m][2];
            //0 meanse short 1 means long
            long ans=(help(s,l,0,0,0,dp,m)+help(s,l,0,0,1,dp,m))%mod;
            System.out.println(ans);
        }
        System.out.println(sb);
    }
    public static long help(long s[],long l[],int idx,int day,int k,Long dp[][][],int m){
        if(day==m-1) return 0;
        if(dp[idx][day][k]!=null) return dp[idx][day][k]%mod;
        long ans=0;
        for(int j=0;j<s.length;j++){
            if(k==1){
                if(s[j]!=0) ans=(ans+s[j]+s[j]+help(s,l,j,day+1,0,dp,m))%mod;
                if(s[j]!=0 && l[j]!=0) ans=(ans+s[j]+l[j]+help(s,l,j,day+1,1,dp,m))%mod;
            }
            else{
                if(s[j]!=0 && l[j]!=0) ans=(ans+s[j]+l[j]+help(s,l,j,day+1,1,dp,m))%mod;
                if(s[j]!=0) ans=(ans+s[j]+s[j]+help(s,l,j,day+1,0,dp,m))%mod;
                if(l[j]!=0) ans=(ans+l[j]+l[j]+help(s,l,j,day+1,1,dp,m))%mod;
                if(s[j]!=0 && l[j]!=0) ans=(ans+l[j]+s[j]+help(s,l,j,day+1,0,dp,m))%mod;
            }
        }
        return dp[idx][day][k]=ans%mod;
    }
    public static long getSum(int r1, int r2, int c1, int c2,long dp[][])
    {
        return dp[r2][c2] - dp[r2][c1] - dp[r1][c2]+ dp[r1][c1];
    }
    public static long inv(long a){
        if(a<=1) return a;
        return mod-(long)(mod/a)*inv(mod%a)%mod;
    }
    public static boolean palin(String s)
    {
        String tmp=s;
        String rev=new StringBuilder(tmp).reverse().toString();
        return s.equals(rev);
    }
    public static long nCr(long n, long r, long p)
    {
        if (r > n - r) r = n - r;
        long C[] = new long[(int)r + 1];
        C[0] = 1;
        for (long i = 1; i <= n; i++) {
            for (long j = Math.min(i, r); j > 0; j--)
                C[(int)j] = (C[(int)j] + C[(int)j - 1]) % p;
        }
        return C[(int)r];
    }
    public static int upper(ArrayList<Integer> arr,int val)
    {
        int low=-1,high=arr.size(),id=-1;
        while(high>1+low)
        {
            int mid=(low+high)/2;
            if(arr.get(mid)>=val) high=mid;
            else low=mid;
        }
        return high;
    }
    public static int lower(ArrayList<Long> arr,long val)
    {
        int low=-1,high=arr.size(),id=-1;
        while(high>1+low)
        {
            int mid=(low+high)/2;
            if(arr.get(mid)<=val) low=mid;
            else high=mid;
        }
        return low;
    }
	 public static long pow(long x, long y,long mod)
    {
        if(y==0)
            return 1;
        if(y==1)
            return x;
        long ans=pow(x,y/2,mod);
        ans=(ans*ans)%mod;
        if(y%2==1)
            ans=(ans*x)%mod;
        return (long)ans;
    }
    private static long gcd(long a, long b) {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }
    private static long lcm(long a, long b) {
        return (a / gcd(a, b)) * b;
    }
	static class FastReader {
        BufferedReader br;
        StringTokenizer st;

        public FastReader()
        {
            br = new BufferedReader(
                new InputStreamReader(System.in));
        }

        String next()
        {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                }
                catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() { return Integer.parseInt(next()); }

        long nextLong() { return Long.parseLong(next()); }

        double nextDouble()
        {
            return Double.parseDouble(next());
        }

        String nextLine()
        {
            String str = "";
            try {
                str = br.readLine();
            }
            catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }
    }
}

**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 edge cases and scenarios that can maximize the failure probability. Here are the steps to think through this:

1. **Edge Cases for Input Constraints:**
   - Minimum values for `m` and `n` (1).
   - Maximum values for `m` and `n` (100 and 1000 respectively).
   - Zero values for `s_i` and `l_i`.

2. **Scenarios to Test:**
   - All short trails (no long trails).
   - All long trails (no short trails).
   - Mixed trails with a combination of short and long trails.
   - A single cabin with multiple trails.
   - Multiple cabins with varying numbers of trails.

3. **Challenging Input:**
   - A small-scale test input that includes a mix of short and long trails, and a reasonable number of cabins and days.

Given these considerations, let's create a test input that includes a mix of short and long trails with a small number of cabins and days. This will help us verify if the code handles the transition between short and long trails correctly.

```plaintext
3 3
1 2 0
0 1 2
```

**Explanation:**
- `m = 3`: There are 3 cabins.
- `n = 3`: Harry walks for 3 days.
- `s = [1, 2, 0]`: Cabin 1 has 1 short trail, Cabin 2 has 2 short trails, and Cabin 3 has 0 short trails.
- `l = [0, 1, 2]`: Cabin 1 has 0 long trails, Cabin 2 has 1 long trail, and Cabin 3 has 2 long trails.

This input includes a mix of short and long trails and a small number of cabins and days, making it a good candidate to expose any bugs in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Easy)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^3$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
import java.util.Scanner;

public class Trails1 {

    public static void main(String[] args) {
        Scanner sc=new Scanner(System.in);
        int n = sc.nextInt();
        int d = sc.nextInt();
        int[][] arr = new int[n][2];
        for (int j = 0; j < 2; j++) {
            for (int i = 0; i < n; i++) {
                arr[i][j] = sc.nextInt();
            }
        }

        int ll = 0, ls = 0, ss = 0, sl = 0, ts = 0, t = 0;
        for (int i = 0; i < n; i++) {
            if (arr[i][1] != 0) {
                ll += arr[i][1];
                ls += arr[i][0];
            }
            if (arr[i][0] != 0) {
                sl += arr[i][1];
                ss += arr[i][0];
            }
            t += arr[i][1] + arr[i][0];
            ts += arr[i][0];
        }
        long ans = 0;

        int gs = arr[0][0], gl = arr[0][1];
        for (int i = 0; i < d; i++) {
            ans = gs * t + gl * ts;
            int ngs = gs * (ss + ls) + gl * (ss);
            int ngl = gl * (sl) + gs * (ll + sl);
            gs = ngs;
            gl = ngl;
        }
        System.out.println(ans);
    }
}

**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 create a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios that might cause the code to fail. Here are the steps to think through this:

1. **Edge Cases**:
   - **Single Cabin**: This is a simple case but important to check.
   - **No Short Trails**: Ensure there are no short trails from any cabin to the lake.
   - **No Long Trails**: Ensure there are no long trails from any cabin to the lake.
   - **All Trails Short**: All trails are short.
   - **All Trails Long**: All trails are long.
   - **Mixed Trails**: A mix of short and long trails, but ensure the constraints are met.

2. **Constraints**:
   - \(0 \le s_i, l_i \le 10^3\)
   - \(1 \le m \le 10^2\)
   - \(1 \le n \le 10^3\)

3. **Failure Scenarios**:
   - The code might fail if it does not handle the modulo operation correctly.
   - The code might fail if it does not correctly update the number of possible combinations for each day.

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

```plaintext
3 2
0 0 1
1 1 0
```

Explanation:
- **m = 3**: There are 3 cabins.
- **n = 2**: Harry walks for 2 days.
- **Short Trails**: Cabin 1 has 0 short trails, Cabin 2 has 0 short trails, Cabin 3 has 1 short trail.
- **Long Trails**: Cabin 1 has 1 long trail, Cabin 2 has 1 long trail, Cabin 3 has 0 long trails.

This test input is designed to:
- Ensure that there are cabins with no short trails.
- Ensure that there are cabins with no long trails.
- Mix the trails in a way that challenges the logic of the code, especially the update of `gs` and `gl` for each day.

This should help expose 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: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
// Problem: C. Light Switches
// Contest: Codeforces - Codeforces Round 963 (Div. 2)
// URL: https://codeforces.com/contest/1993/problem/C
// Memory Limit: 256 MB
// Time Limit: 2000 ms
//
// Powered by CP Editor (https://cpeditor.org)

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()
using namespace std;
using ll = long long;
using ull = unsigned long long;

const int _N = 1e5 + 5;

int T;

void solve() {
	ll n, k; cin >> n >> k;
	vector<ll> a(n + 1);
	for (int i = 1; i <= n; i++) cin >> a[i];
	ll l = a[1], r = a[1] + k - 1;
	for (int i = 2; i <= n; i++) {
		ll x = a[i], y = a[i] + k - 1;
		if (x == l) continue;
		// [x, y], [l (+2k * n), r (+2k * n)]
		if (l < x) {
			swap(l, x);
			swap(r, y);
		}
		ll m = (l - x) / (2 * k);
		if (y + 2 * k * m >= l) r = min(r, y + 2 * k * m);
		else if (x + 2 * k * (m + 1) <= r) l = max(l, x + 2 * k * (m + 1));
		else {
			cout << -1 << '\n';
			return;
		}
	}
	cout << l << '\n';
	return;
}

int main() {
	ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	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
4 2
1 3 5 7
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define ll long long
void solve(){
   int n ,k;
    cin>>n>>k;
    vector<int> a(n);
    for(int i=0;i<n;i++){
        cin>>a[i];
    }
    sort(a.begin(),a.end());
    pair<int,int> ans;
    ans.first=a[n-1];
    ans.second=a[n-1]+k;
    for(int i=n-1;i>=0;i--){
        int ele = (ans.first-a[i])/k;
        if(ele%2==0)
        ans.second=a[i]+ele*k+k;
        else
        {
            if(a[i]+ele*k==ans.first)
            {
                cout<<-1<<"\n";
                return;
            }else
            ans.first=a[i]+ele*k+k;
        }
        if(ans.first>ans.second)
        {
            cout<<-1<<"\n";
            return;
        }
    }
        //cout<<ans.first<<" "<<ans.second<<"\n";
        if(ans.first<=ans.second)
    cout<<ans.first<<"\n";
    else
    cout<<-1<<"\n";
    }

int main(){
    ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
    int t;
    cin>>t;
    while(t--){
        solve();
    }
    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
2
4 2
1 3 5 7
3 3
1 4 7
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
#include <iostream>
#include <fstream>
#include <string>
#include <array>
#include <vector>
#include <algorithm>
#include <limits>
#include <set>
#include <math.h>
#include <queue>
#include <iomanip>
#include <map>
#include <cstring>
#include <numeric>
#include <cstdio>
#include <bitset>
#include <cstdint>
#include <assert.h>
#include <limits.h>
#include <unordered_map>
#include <memory>
#include <istream>

using ll = long long;

using namespace std;

int main() {
	ios_base::sync_with_stdio(false);
	cin.tie(NULL);
	//ifstream cin("a.in");
	int t;
	cin >> t;
	while (t--) {
		int n, k;
		bool no = 0;
		cin >> n >> k;
		vector<ll> a(n);
		for (int i = 0; i < n; i++)
			cin >> a[i];
		sort(a.begin(), a.end());
		ll l = 0, r = 2 * k - 1, kmult;
		for (int i = 0; i < n; i++) {
			ll cur = a[i] % (2 * k);
			kmult = a[i] / (2 * k);
			if (cur > r) {
				cur -= (2 * k);
				kmult++;
			}
			if (cur + k - 1 < l) {
				no = 1;
				break;
			}
			l = max(l, cur);
			r = min(r, cur + k - 1);
		}
		if (no) {
			cout << -1 << endl;
			continue;
		}
		cout << l + (2 * k) * kmult << endl;
	}
}

**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
4 4
1 5 9 13
3 3
1 2 3
2 1
1 1000000000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
from sys import stdin
def input(): return stdin.readline()[:-1]
def ili(): return list(map(int, input().split()))
def fastout(S): print(' '.join(map(str, S)))

def solve():
    n, k = map(int, input().split())
    a = ili()
    maxxr = 0
    for i, ele in enumerate(a):
        maxxr = max(maxxr, ele//(2*k))
        a[i] %= (2*k)
    # print(maxxr)
    # print(a)
    # mx = max(a)
    if (max(a) - min(a)) < k:
        print(maxxr*2*k + max(a))
        return
    else:
        for i in range(len(a)):
            a[i] = (a[i] + k) % (2*k)
        if (max(a) - min(a)) < k:
            print(maxxr*2*k + k + max(a))
            return
    print(-1)
    return

case_num = 1
case_num = int(input())
for case in range(case_num):
    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
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: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
import sys
input=sys.stdin.readline

for _ in range(int(input())):
    n, k = map(int, input().split())
    ar = list(map(int, input().split()))
    ans = 0
    l = ar[0] % (2 * k)
    r = l + k
    for i in range(n):
        x = ar[i] // (2 * k)
        y = ar[i] % (2 * k)
        if l <= y < r:
            l = y
        elif y < l and y + k > l:
            r = y + k
        elif y >= r and (y + k) % (2 * k) > l:
            x += 1
            r = (y + k) % (2 * k)
        else:
            ans = -1
            break
        if x > ans:
            ans = x
    if ans >= 0:
        print(ans * 2 * k + l)
    else:
        print(-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
4 4
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: Light Switches
time_limit_ms: 2000
memory_limit_mb: 256
Description: There is an apartment consisting of $n$ rooms, each with its light initially turned off.

To control the lights in these rooms, the owner of the apartment decided to install chips in the rooms so that each room has exactly one chip, and the chips are installed at different times. Specifically, these times are represented by the array $a_1, a_2, \ldots, a_n$, where $a_i$ is the time (in minutes) at which a chip is installed in the $i$-th room.

As soon as a chip is installed, it changes the room's light status every $k$ minutes — it turns on the light for $k$ minutes, then turns it off for the next $k$ minutes, then turns it back on for the next $k$ minutes, and so on. In other words, the light status is changed by the chip at minute $a_i$, $a_i + k$, $a_i + 2k$, $a_i + 3k$, $\ldots$ for the $i$-th room.

What is the earliest moment when all rooms in the apartment have their lights turned on?

Input: The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 2 \cdot 10^5$) — the number of rooms in the apartment and the period of the chips.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the moments when the chips are installed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, print a single integer — the answer to the question (in minutes). If there is no such moment that the lights are turned on in all the rooms, print $-1$ instead.

Examples:
input:
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
output:
5
-1
10
8
1
47
100
-1
-1

Note: In the first test case, all lights will be on by the minute $5$ without any of them being turned off by the chips. The answer is $5$.

In the second test case, due to $k=3$, the $1$-st light will be on at minutes $2, 3, 4, 8, 9, 10, 14, \ldots$; meanwhile, the $4$-th light will be on at minutes $5, 6, 7, 11, 12, 13, 17, \ldots$. These two sequences don't have any number in common, so they will never be on at the same time.

In the third test case, it can be seen that the $1$-st and $2$-nd lights will be turned off at minutes $6$ and $7$, but the chips will turn them back on at minutes $9$ and $10$. The $3$-rd and $4$-th lights will also be on at minute $10$, so the answer is $10$.

Buggy Code:
def main():
    for _ in range(int(input())):
        time = 0
        n, k = map(int, input().split())
        time_list = [(int(_), False) for _ in input().split()]
        while time < pow(10, 5):
            time_list = [(i + k, not j) if i <= time else (i, j) for (i, j) in time_list]
            if all(j for (_, j) in time_list):
                print(time)
                break
            time += 1
        else:
            print(-1)

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:
```plaintext
9
4 4
2 3 4 5
4 3
2 3 4 5
4 3
3 4 8 9
3 3
6 2 1
1 1
1
7 5
14 34 6 25 46 7 17
6 5
40 80 99 60 90 50
6 5
64 40 50 68 70 10
2 1
1 1000000000
```