**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Monster
time_limit_ms: 2000
memory_limit_mb: 256
Description: Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

  * Increase $d$ — the damage of your weapon by $1$, costing $x$ coins.    * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

Input: The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

Output: For each test case, output the minimum number of coins needed to defeat the monster.

Examples:
input:
4
2 3 5 5
10 20 40 5
1 60 100 10
60 1 100 10
output:
12
190
280
160

Note: In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

  * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

  * Increase damage by $5$, costing $5\cdot x$ = $50$ coins.    * Attack the monster once, dealing $5$ damage, costing $20$ coins.    * Increase damage by $2$, costing $2\cdot x$ = $20$ coins.    * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal a total of $5 + 35 = 40$ damage, defeating the monster who has exactly $40$ health. The total number of coins you use is $50 + 20 + 20 + 100 = 190$ coins.

Buggy Code:
import sys
import collections
import math
import random
from functools import lru_cache

I = lambda : [*map(int, sys.stdin.readline().split())]

def solve(t):
    x, y, z, k = I()

    # disc = (k**6 * x**2  + 8 * k**5 * x**2 * z + 2 * k**3 *x*y + 16*k**2*x*y*z) ** 0.5
    disc = (2*k**3*x*y + 16*k**2*x*y*z + k**2*y**2 + 8*k*y**2*z) ** 0.5
    approx1 = -2 * k**2 * x - disc - k*y
    approx1 /= 2*(2*k**2*x + k*y)
    approx1 = int(approx1)

    approx2 = -2 * k**2 * x + disc - k*y
    approx2 /= 2*(2*k**2*x + k*y)
    approx2 = int(approx1)

    # print(approx1,approx2)

    already = 10**18
    for i in range(1,min(5*10**5,k+1)):
        if i <= z:
            already = min(already, i*x + (z + i - 1) // i * y)

    chk = [approx1,approx2]
    best = 10**18
    for approx in chk:
        chkp = [0]
        # p*(x*k + y) <= already
        for p in range(max(0, approx - 50000), min(int((2*z/k)**0.5 + 5), already // (x*k+y) + 2, approx + 50000)):
            chkp.append(p)
        for p in chkp:
            part1_damage = k*p*(p+1) // 2
            part1_cost = p*(x*k + y)
            if part1_damage >= z:
                best = min(best, part1_cost)
                continue

            approx_e = (y*(z - k*p*(p+1)//2) / x) ** 0.5 - p*k
            approx_e = int(approx_e)

            for ee in range(max(0, approx_e - 75000), min(k, approx_e + 75000)):
                denom = p*k+ee
                if denom == 0:
                    continue
                part2_cost = ee*x + (z - k*p*(p+1)//2 + denom-1) // denom * y

                best = min(best, part1_cost + part2_cost)

    print(best)

tt = 1
tt = I()[0]
for _ in range(tt):
    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
1 1 100000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Monster
time_limit_ms: 2000
memory_limit_mb: 256
Description: Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

  * Increase $d$ — the damage of your weapon by $1$, costing $x$ coins.    * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

Input: The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

Output: For each test case, output the minimum number of coins needed to defeat the monster.

Examples:
input:
4
2 3 5 5
10 20 40 5
1 60 100 10
60 1 100 10
output:
12
190
280
160

Note: In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

  * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

  * Increase damage by $5$, costing $5\cdot x$ = $50$ coins.    * Attack the monster once, dealing $5$ damage, costing $20$ coins.    * Increase damage by $2$, costing $2\cdot x$ = $20$ coins.    * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal a total of $5 + 35 = 40$ damage, defeating the monster who has exactly $40$ health. The total number of coins you use is $50 + 20 + 20 + 100 = 190$ coins.

Buggy Code:
# standard inputs
import itertools
from collections import *
from bisect import *
import heapq
from functools import *
import math
import sys
import random
from copy import deepcopy
#from templates import *; fin=open('i', 'r'); input = lambda: fin.readline().strip(); from icecream import ic; from sortedcontainers import SortedList
from fractions import Fraction

R = lambda: [*map(int, input().split())]; # read one line(turn to int array)
lines = lambda x: [R() for i in range(x)]; # read multiple input lines

M = int(1e9+7)

def isMtn(A):
  if len(A) <= 2:
    return True
  for i in range(len(A)+1):
    a = A[:i]
    b = A[i:]
    if a == sorted(a, reverse=True) and b == sorted(b):
      return i
  raise

CHECK_LEN = 100000
EPS = 1e-7
def solve(x, y, z, k):
  #ic(x, y, z, k)
  def calc2(A, mode='fake'):
    if A <= 0:
      return math.inf
    return calc(A//(k+1), A%(k+1), mode)
  # if A is #blocks
  def calc(A, rem, mode='fake'):
    if A == 0 and rem == 0:
      return math.inf
    if rem > k:
      return math.inf

    blocksum =  y*A + x*k*A
    remsum = x*rem
    dmg = k*A + rem

    health = z-k*A*(A+1)//2
    delsum = (math.ceil(health/dmg) if mode == 'real' else health/dmg) * y
    return blocksum+remsum+delsum

  def bsearch(l, r, fn):
    #ic([fn(i) for i in range(l, r+1)])
    while l+1 < r:
      mid = (l+r)//2
      left_eq, right_eq = mid, mid

      while left_eq > 0 and abs(fn(left_eq) - fn(mid)) < EPS:
        left_eq -= 1
      while left_eq > 0 and abs(fn(right_eq) - fn(mid)) < EPS:
        right_eq += 1

      if fn(left_eq) < fn(mid):
        r = left_eq+1
      elif fn(right_eq) < fn(mid):
        l = right_eq-1
      else:
        return mid
    #ic(l)
    return max(l-2, l-1, l, l+1, l+2, key=calc2)

  def getAns(blocks):
    if blocks < 0:
      return math.inf
    rem = bsearch(blocks==0, k, lambda rem: calc(blocks, rem, 'fake'))
    return calc(blocks, rem)

  inc = bsearch(1, z, calc2)
  ans = 1e100
  for i in range(max(1, inc-CHECK_LEN), inc+CHECK_LEN):
    ans = min(ans, calc2(i, mode='real'))

  return ans

for _ in range(int(input())):
  x, y, z, k = R()
  print(solve(x, y, z, k))

#for _ in range(100):
#  x, y, z, k = [random.randint(1, int(1e8))] * 4
#  print(solve(x, y, z, k))

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

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

Think step by step.

Response:
```plaintext
3
1 1 1 1
1 1 2 1
1 1 100000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Monster
time_limit_ms: 2000
memory_limit_mb: 256
Description: Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

  * Increase $d$ — the damage of your weapon by $1$, costing $x$ coins.    * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

Input: The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

Output: For each test case, output the minimum number of coins needed to defeat the monster.

Examples:
input:
4
2 3 5 5
10 20 40 5
1 60 100 10
60 1 100 10
output:
12
190
280
160

Note: In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

  * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

  * Increase damage by $5$, costing $5\cdot x$ = $50$ coins.    * Attack the monster once, dealing $5$ damage, costing $20$ coins.    * Increase damage by $2$, costing $2\cdot x$ = $20$ coins.    * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal a total of $5 + 35 = 40$ damage, defeating the monster who has exactly $40$ health. The total number of coins you use is $50 + 20 + 20 + 100 = 190$ coins.

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
public class Codeforce2035E {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        String[] inputs = br.lines().toArray(String[]::new);
        for (int i = 1; i < inputs.length; i++) {
            String[] split = inputs[i].split(" ");
            long x = Long.parseLong(split[0]);
            long y = Long.parseLong(split[1]);
            long z = Long.parseLong(split[2]);
            long k = Long.parseLong(split[3]);
            System.out.println(solve(x, y, z, k));
        }
        br.close();
    }
    private static long solve(long x, long y, long z, long k) {
        long result = Long.MAX_VALUE;
        for (long i = 1;i<30000 ; i++) {
            boolean breakTag = false;
            long remainder = i / k;
            long ladderSum = (k + remainder * k) * remainder / 2;
            if(ladderSum > z){
                result = Math.min(result, i * x + remainder * y);
                breakTag = true;
            } else {
                result = Math.min(result, i * x + remainder * y + ((long) Math.ceil((double) (z - ladderSum) / i)) * y) ;
            }
            if(breakTag){
                break;
            }
        }
        return result;
    }
}

**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
1 1 100000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Monster
time_limit_ms: 2000
memory_limit_mb: 256
Description: Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

  * Increase $d$ — the damage of your weapon by $1$, costing $x$ coins.    * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

Input: The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

Output: For each test case, output the minimum number of coins needed to defeat the monster.

Examples:
input:
4
2 3 5 5
10 20 40 5
1 60 100 10
60 1 100 10
output:
12
190
280
160

Note: In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

  * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Increase damage by $1$, costing $2$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.    * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

  * Increase damage by $5$, costing $5\cdot x$ = $50$ coins.    * Attack the monster once, dealing $5$ damage, costing $20$ coins.    * Increase damage by $2$, costing $2\cdot x$ = $20$ coins.    * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal a total of $5 + 35 = 40$ damage, defeating the monster who has exactly $40$ health. The total number of coins you use is $50 + 20 + 20 + 100 = 190$ coins.

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

public class Main {
    static PrintWriter pw = new PrintWriter(System.out);
    static Scanner sc = new Scanner(System.in);

    public static long sumMultiplesOfK(long k, long n) {
        long m = n / k;
        return k * m * (m + 1) / 2;
    }

    static long ans;

//    public static long ternarySearch(long low, long high, long x, long y, long z, long k) {
//        long mid1 = low + (high - low) / 3;
//        long mid2 = high - (high - low) / 3;
//
//        long cnt1 = mid1 * x;
//        long ans1 = z - sumMultiplesOfK(k, mid1);
//        cnt1 += (mid1 / k) * y;
//        cnt1 += ((ans1 + mid1 - 1) / mid1) * y;
//
//
//        long cnt2 = mid2 * x;
//        long ans2 = z - sumMultiplesOfK(k, mid2);
//        cnt2 += (mid2 / k) * y;
//        cnt2 += ((ans2 + mid2 - 1) / mid2) * y;
//
//        if(cnt1 < cnt2) {
//            high = mid2 - 1;
//            ans = Math.min(ans, cnt1);
//            return ternarySearch(low, mid1 - 1, x, y, z, k);
//        }
//        else{
//            low = mid1 + 1;
//            ans = Math.min(ans, cnt2);
//            return ternarySearch(mid2 + 1, high, x, y, z, k);
//        }
//
//    }

    static class Pair implements Comparable<Pair>{
        Long x ;
        Long y ;

        public 	Pair(Long i, Long j) {
            this.x = i;
            this.y = j;
        }
        public int compareTo(Pair p) {
            if (this.x.compareTo(p.x) != 0)
                return -this.x.compareTo(p.x);
            else
                return this.y.compareTo(p.y);
        }

        public String toString(){
            return "("+this.x + "," + this.y + ")";
        }
    }

    public static void solve() throws IOException {
        long x = sc.nextInt(), y = sc.nextInt(), z = sc.nextInt(), k = sc.nextInt();
        ans = Long.MAX_VALUE;
        if(z <= 1000){
            for(int i = 1; i <= z; i++){
                long cur = i * x;
                long rem = z - sumMultiplesOfK(k, i);
                cur += (i / k) * y;
                cur += ((rem + i - 1) / i) * y;
                ans = Math.min(ans, cur);
            }
            pw.println(ans);
            return;
        }
        long cur1 = -1, cur2 = -1;
        int idx1 = -1, idx2 = -1;
        for(int i = 1 ; i <= z ; i+= 250){
            long cur = i * x;
            long rem = z - sumMultiplesOfK(k, i);
            cur += (i / k) * y;
            cur += ((rem + i - 1) / i) * y;
            if(cur1 == -1){
                cur1 = cur;
                idx1 = i;
            }
            else if(cur2 == -1){
                cur2 = cur;
                idx2 = i;
            }
            else{
                if(cur <= cur2){
                    cur1 = cur2;
                    idx1 = idx2;
                    cur2 = cur;
                    idx2 = i;
                }
            }
        }
        for(int i = Math.max(1, idx1 - 1000); i <= Math.min(z, idx2 + 1000); i++){
            long cur = i * x;
            long rem = z - sumMultiplesOfK(k, i);
            cur += (i / k) * y;
            cur += ((rem + i - 1) / i) * y;
            ans = Math.min(ans, cur);
        }
        pw.println(ans);
    }

    public static void main(String[] args) throws IOException {
        int t = sc.nextInt();
        while(t -- > 0)
            solve();
        pw.flush();
    }

    static class Scanner {
        StringTokenizer st;
        BufferedReader br;

        public Scanner(InputStream s) {
            br = new BufferedReader(new InputStreamReader(s));
        }

        public Scanner(String s) throws FileNotFoundException {
            br = new BufferedReader(new FileReader(s));
        }

        public String next() throws IOException {
            while (st == null || !st.hasMoreTokens())
                st = new StringTokenizer(br.readLine());
            return st.nextToken();
        }

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

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

        public String nextLine() throws IOException {
            return br.readLine();
        }

        public double nextDouble() throws IOException {
            String x = next();
            StringBuilder sb = new StringBuilder("0");
            double res = 0, f = 1;
            boolean dec = false, neg = false;
            int start = 0;
            if (x.charAt(0) == '-') {
                neg = true;
                start++;
            }
            for (int i = start; i < x.length(); i++)
                if (x.charAt(i) == '.') {
                    res = Long.parseLong(sb.toString());
                    sb = new StringBuilder("0");
                    dec = true;
                } else {
                    sb.append(x.charAt(i));
                    if (dec)
                        f *= 10;
                }
            res += Long.parseLong(sb.toString()) / f;
            return res * (neg ? -1 : 1);
        }

        public long[] nextlongArray(int n) throws IOException {
            long[] a = new long[n];
            for (int i = 0; i < n; i++)
                a[i] = nextLong();
            return a;
        }

        public Long[] nextLongArray(int n) throws IOException {
            Long[] a = new Long[n];
            for (int i = 0; i < n; i++)
                a[i] = nextLong();
            return a;
        }

        public int[] nextIntArray(int n) throws IOException {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt();
            return a;
        }

        public Integer[] nextIntegerArray(int n) throws IOException {
            Integer[] a = new Integer[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt();
            return a;
        }

        public boolean ready() throws IOException {
            return br.ready();
        }
    }
}

**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
1 100000000 100000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

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

int main(){
	int t;
	cin>>t;
	while(t--){
		int n;
		cin>>n;
		vector<int> a(n);
		set<int> s;
		bool ans = true;
		for(int i = 0; i<n; i++){
			cin>>a[i];
			s.insert(a[i]);
		}
		int m;
		cin>>m;
		int sets = s.size();
		if(sets>26){
			ans = false;
		}
		for(int i = 0; i<m; i++){
			string p;
			cin>>p;
			// cout<<p<<endl;
			if(ans == false){
				cout<<"NO"<<endl;
			}
			else{

				int l = p.size();
				if(l!=n){
					cout<<"NO"<<endl;
				}
				else{
					int ans2 = true;
					map<char, int> mp;
					for(int j = 0; j<n; j++){
						if(mp.find(p[j]) == mp.end()){
							mp[p[j]] = a[j];
						}
						else{
							if(mp[p[j]] != a[j]){
								ans2 = false;
								break;
							}
						}
					}
					if(ans2 == true){
						cout<<"YES"<<endl;
					}
					else{
						cout<<"NO"<<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
1
5
3 5 2 1 3
3
abfda
afbfa
abfca
```

### Explanation:
1. **Test Case 1:**
   - **Array `a`:** [3, 5, 2, 1, 3]
   - **Number of Strings `m`:** 3
   - **Strings:**
     - "abfda"
     - "afbfa"
     - "abfca"

2. **Expected Output:**
   - For "abfda": **YES** (matches the template)
   - For "afbfa": **NO** (does not match the template because 'f' corresponds to both 1 and 5)
   - For "abfca": **NO** (does not match the template because 'c' and 'a' correspond to different values in `a`)

### Rationale:
- The first string "abfda" is a valid match because it satisfies the one-to-one correspondence between the characters and the numbers in the array.
- The second string "afbfa" is invalid because the character 'f' corresponds to both 1 and 5, violating the one-to-one correspondence rule.
- The third string "abfca" is invalid because the character 'c' corresponds to a different number than 'a', which is already mapped to 3.

This test case is designed to expose the bug in the given code where the one-to-one correspondence between characters and numbers in the array is not correctly checked. The code might incorrectly pass the second string "afbfa" as a match, which it should not.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
using i64 = long long;
using u32 = unsigned;
using u64 = unsigned long long;

void solve() {
    int n; cin >> n;
    vector<i64> a(n);
    for (i64 &i: a) cin >> i;
    int m; cin >> m;
    while (m--) {
        map<char,i64> b;
        string s; cin >> s;
        bool flag = true;
        if (s.length()!=a.size()) {
            cout << "NO\n";
            continue;
        }
        for (int i=0;i<n;++i) {
            if (b.count(s[i])==0) {
                b[s[i]] = a[i];
            } else {
                if (b[s[i]]!=a[i]) {
                    flag = false;
                    break;
                }
            }
        }
        if (flag) cout << "YES\n";
        else cout << "NO\n";
    }
}

int main() {
    ios::sync_with_stdio(false); cin.tie(nullptr);
    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
3
5
3 5 2 1 3
2
abfda
afbfa
4
1 2 1 2
4
abab
aabb
abba
baba
3
1 1 1
3
aaa
abc
aba
```

3
5
3 5 2 1 3
2
abfda
afbfa
4
1 2 1 2
4
abab
aabb
abba
baba
3
1 1 1
3
aaa
abc
aba


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
int main()
{
    long long int t;
    cin>>t;
    while(t--)
    {
        long long int n;
        cin>>n;
        long long int a[n];
        for(auto &it:a) cin>>it;
        long long int m;
        cin>>m;
        string str[m];
        long long int ans[m];
        memset(ans,0,sizeof(ans));
        for(auto &it:str) cin>>it;
        for(long long int i=0;i<m;i++)
        {
            string s=str[i];
            if(s.length()!=n)
            {
                ans[i]=0;
                continue;
            }
            bool b=true;
            map<char,long long int> m;
            for(int j=0;j<n;j++)
            {
                if(a[j]==0)
                {
                    a[j]=1e9+10;
                }
                if(m[s[j]]==0)
                {
                    m[s[j]]=a[j];
                }
                else
                {
                    if(m[s[j]]==a[j])
                    {
                        continue;
                    }
                    else
                    {
                        b=false;
                        break;
                    }
                }
            }
            if(b)
            {
                ans[i]=1;
            }
            else
            {
                ans[i]=0;
            }
        }
        for(long long int i=0;i<m;i++)
        {
            if(ans[i])
            {
                cout<<"YES"<<endl;
            }
            else
            {
                cout<<"NO"<<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
1
5
1 2 3 4 5
2
abcde
abcdf
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
t= int(input())
anws = []
for _ in range(t):
    n = int(input())
    a = list(map(int, input().split()))
    m = int(input())
    for __ in range(m):
        s = input()
        # 1st condition
        if n != len(s):
            anws.append("NO")
            continue
        # 2nd condition
        logic = True
        a_dict = dict()
        for i in a:
            a_dict[i] = 0
        for i in a:
            a_dict[i] +=1
        for i in a_dict:
            if a_dict[i] > 1:
                indexes = []
                for z,x in enumerate(a):
                    if x == i:
                        indexes.append(z)
                first_element_of_s= s[indexes[0]]
                for x in indexes:
                    if s[x] != first_element_of_s:
                        anws.append("NO")
                        logic = False
                        break

                #collect indexes of i and check it with s
        # 3rd condition
        if logic == True:
            s_dict = dict()
            for i in s:
                s_dict[i] = 0
            for i in s:
                s_dict[i] +=1
            for i in s_dict:
                if s_dict[i] > 1:
                    indexes = []
                    for z,x in enumerate(s):
                        if x == i:
                            indexes.append(z)
                    first_element_of_a= a[indexes[0]]
                    for x in indexes:
                        if a[x] != first_element_of_a:
                            anws.append("NO")
                            logic = False

                            break
        if logic == True:
            anws.append("YES")
for i in anws:
    print(i)

**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 highlight the issues in the logic. The buggy code has a few potential issues, such as incorrect handling of one-to-one correspondence between characters and numbers, and potential off-by-one errors in index checking.

Let's consider the following scenarios:

1. **Strings with repeated characters and numbers**: This can test the one-to-one correspondence logic.
2. **Strings with different lengths**: This can test the length condition.
3. **Strings with unique characters and numbers**: This can test the one-to-one correspondence logic in a simpler case.
4. **Empty strings and arrays**: This can test edge cases where the input is minimal or invalid.

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

```plaintext
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
```

This input includes:
1. A template with repeated numbers and a string that should match and one that should not.
2. A template with unique numbers and strings of different lengths.
3. A template with repeated numbers and strings that should match and one that should not.

This input covers a variety of edge cases and should help expose any issues in the buggy code.

3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
for _ in range(int(input())):
    n  = int(input())
    a = list(map(int,input().split()))
    m = int(input())

    for i in range(m):
        d = {}
        ans = 'yes'
        s = input()
        if len(s) == len(a):
            for i in range(n):
                if s[i] not in d.keys():
                    d[s[i]] = []
                    d[s[i]].append(a[i])
                else:
                    if d[s[i]][0]!=a[i] or len(d[s[i]])!=1:
                        ans='no'
        else:
            ans='no'
        # print(d)
        print(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 scenarios where the code might fail. The primary issues in the buggy code are:

1. The code does not correctly handle the case where multiple characters map to the same number.
2. The code does not correctly handle the case where the same character maps to different numbers.
3. The code does not correctly handle the case where the lengths of the string and the array do not match.

Let's create a small-scale test input that covers these scenarios:

1. **Different characters mapping to the same number:**
   - Array: [1, 1, 2, 2]
   - String: "aabb" (should be "NO" because 'a' and 'b' both map to 1 and 2)
   - String: "abab" (should be "YES" because 'a' maps to 1 and 'b' maps to 2)

2. **Same character mapping to different numbers:**
   - Array: [1, 2, 1, 2]
   - String: "abab" (should be "NO" because 'a' maps to both 1 and 2)

3. **Length mismatch:**
   - Array: [1, 2, 3]
   - String: "abc" (should be "YES")
   - String: "ab" (should be "NO" because the lengths do not match)

Here is the test input:

```plaintext
3
4
1 1 2 2
2
aabb
abab
4
1 2 1 2
1
abab
3
1 2 3
2
abc
ab
```

This test input includes:
- A case where different characters map to the same number.
- A case where the same character maps to different numbers.
- A case where the lengths of the string and the array do not match.

These scenarios should help expose the bugs in the provided buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
s=int(input())
for _ in range(s):
    n=int(input())
    arr=input()
    no_spaces =arr.replace(" ", "")
    m=int(input())
    ch1=True
    for j in range(m):
            ch=input()
            if len(ch)!=n:
                ch1=False

            else:
                for j in range(len(ch)):
                    for i in range(len(no_spaces)):
                        if no_spaces.count(no_spaces[i])!=ch.count(ch[j]) and i==j and no_spaces.index(no_spaces[i])!=ch.index(ch[j]):
                           ch1=False

            if ch1:
                print('yes')

            else:
                print('no')
                ch1=True

**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 given buggy code, let's consider the following steps:

1. **Identify the Bug**: The buggy code has a logical flaw in how it checks the one-to-one correspondence between the characters of the string and the elements of the array. It incorrectly uses nested loops and conditions that do not correctly validate the one-to-one correspondence.

2. **Create Test Cases**:
   - **Test Case 1**: A simple case where the template and the string match.
   - **Test Case 2**: A case where the template and the string do not match due to the same characters corresponding to different numbers.
   - **Test Case 3**: A case where the template and the string do not match due to different characters corresponding to the same number.
   - **Test Case 4**: A case with a large number of strings to check the performance and correctness of the code.

3. **Edge Cases**:
   - **Edge Case 1**: An empty string or a string with a single character.
   - **Edge Case 2**: A template with repeated numbers and strings with repeated characters.
   - **Edge Case 3**: A template with unique numbers and strings with unique characters.

Given these considerations, let's generate a small-scale test input that includes these cases:

```plaintext
4
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
3
1 1 1
3
aaa
aba
abc
3
1 2 3
3
abc
aba
aaa
```

### Explanation of the Test Cases:

1. **Test Case 1**:
   - Template: `3 5 2 1 3`
   - String 1: `abfda` (matches the template)
   - String 2: `afbfa` (does not match the template)

2. **Test Case 2**:
   - Template: `1 2`
   - String 1: `ab` (matches the template)
   - String 2: `abc` (does not match the template due to length)
   - String 3: `aa` (does not match the template due to the same character corresponding to different numbers)

3. **Test Case 3**:
   - Template: `1 1 1`
   - String 1: `aaa` (matches the template)
   - String 2: `aba` (does not match the template)
   - String 3: `abc` (does not match the template)

4. **Test Case 4**:
   - Template: `1 2 3`
   - String 1: `abc` (matches the template)
   - String 2: `aba` (does not match the template)
   - String 3: `aaa` (does not match the template)

This test input should effectively expose the bug in the given buggy code by challenging the logic of one-to-one correspondence and edge cases.

4
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
3
1 1 1
3
aaa
aba
abc
3
1 2 3
3
abc
aba
aaa


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
//package CF_966;

import java.util.*;

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

        int t = sc.nextInt();

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

            int[] arr = new int[n];
            for(int i=0;i<n;i++) arr[i] = sc.nextInt();

            int m = sc.nextInt();
            sc.nextLine();
            for(int i=0;i<m;i++){
                String s = sc.nextLine();
                boolean flag = true;
                if(s.length()!=n){
                    System.out.println("NO");
                    continue;
                }
                HashMap<Character,Integer> hm = new HashMap<>();

                for(int j=0;j<n;j++){
                    char c = s.charAt(j);
                    if(!hm.containsKey(c)) hm.put(c,arr[j]);
                    else{
                        if(hm.get(c)!=arr[j]) flag = false;
                    }
                }
                if(flag) System.out.println("YES");
                else System.out.println("NO");
            }
        }
    }
}

**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
1 2 3 4 5
5
abcde
abced
abcda
abcdd
abcde
```

1
5
1 2 3 4 5
5
abcde
abced
abcda
abcdd
abcde


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
import java.util.HashMap;
import java.util.Map;
import java.util.Scanner;

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

        int T = sc.nextInt();

        while (T-- > 0) {
            solve();
        }
    }

    public static void solve() {
        int n  = sc.nextInt();
        int [] arr = new int[n];
        for (int i = 0 ; i<n ; i++) {
            arr[i] = sc.nextInt();
        }

        int numStr = sc.nextInt();

        for (int i = 0 ; i < numStr ; i++) {
            String str = sc.next();
            boolean correct = true;
            if (str.length() != n) {
                System.out.println("NO");
                continue;
            }

            Map<Character, Integer> map = new HashMap<>();

            for (int j = 0 ; j < str.length() ; j++) {
                if (!map.containsKey(str.charAt(j))) {
                    map.put(str.charAt(j), arr[j]);
                    continue;
                }

                if (map.get(str.charAt(j)) != arr[j]){
                    correct = false;
                    break;
                }
            }
            if (!correct) {
                System.out.println("NO");
            }
            else {
                System.out.println("YES");
            }
        }
    }
}

**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 focus on scenarios where the one-to-one correspondence between the characters of the string and the elements of the array is not correctly handled. The code currently only checks if the same character in the string maps to the same element in the array. However, it does not check if the same element in the array maps to the same character in the string. This can lead to incorrect results.

Here's a step-by-step approach to generate a test input:

1. **Test Case with Repeated Elements in the Array and Different Characters in the String:**
   - The array should have repeated elements.
   - The string should have different characters at the positions of the repeated elements in the array.

2. **Test Case with Repeated Characters in the String and Different Elements in the Array:**
   - The string should have repeated characters.
   - The array should have different elements at the positions of the repeated characters in the string.

3. **Test Case with a Single Element and a Single Character:**
   - This is a simple case to ensure the code handles single-element arrays and single-character strings correctly.

4. **Test Case with Large Input:**
   - This can help test the performance and correctness of the code with larger inputs.

Let's generate a small-scale test input that covers these scenarios:

```plaintext
4
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
1
1
1
a
```

### Explanation:
1. **First Test Case:**
   - Array: [3, 5, 2, 1, 3]
   - Strings: ["abfda", "afbfa"]
   - "abfda" should match (YES)
   - "afbfa" should not match (NO) because 'f' maps to both 1 and 5.

2. **Second Test Case:**
   - Array: [1, 2]
   - Strings: ["ab", "abc", "aa"]
   - "ab" should match (YES)
   - "abc" should not match (NO) because the length does not match.
   - "aa" should not match (NO) because 'a' maps to both 1 and 2.

3. **Third Test Case:**
   - Array: [5, -3, 5, -3]
   - Strings: ["aaaa", "bcbc", "aba", "cbcb"]
   - "aaaa" should not match (NO) because 'a' maps to both 5 and -3.
   - "bcbc" should match (YES)
   - "aba" should not match (NO) because the length does not match.
   - "cbcb" should match (YES)

4. **Fourth Test Case:**
   - Array: [1]
   - String: ["a"]
   - "a" should match (YES)

This test input should effectively expose the bug in the provided code.

4
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
1
1
1
a


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
#include<bits/stdc++.h>
//#include<ext/pb_ds/assoc_container.hpp>
using namespace std;
//using namespace __gnu_pbds;
#define int long long
#define endl '\n'
template <class T> using matrix = vector<vector<T>>;
//template<class C> using oset = tree<C,null_type,less<C>,rb_tree_tag,tree_order_statistics_node_update>;

void solve(){
	int n , m , temp;
	cin>>n;
	map<int , vector<int>> idx;
	for(int i=0; i<n; i++){
		cin>>temp;
		idx[temp].push_back(i);
	}
	cin>>m;
	while(m--){
		int ans = 0 , t;
		string s;
		cin>>s;
		t = (int)s.size();
		set a(s.begin() , s.end());
		if(a.size() != idx.size()){
			ans = 1;
			cout<<"NO\n";
		}

		if(ans) continue;

		for(auto [x, y] : idx){
			if(y[0] >= t) continue;
			char c = s[y[0]];
			for(auto i : y){
				if(i >= t || s[i] != c){
					ans = 1;
					cout<<"NO\n";
					break;
				}
			}
			if(ans) break;
		}
		if(!ans) cout<<"YES\n";
	}
}

int32_t main(){
	ios::sync_with_stdio(0);
	cin.tie(0);

	//freopen("ride.in", "r", stdin);
	//freopen("ride.out", "w", stdout);

	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 given buggy code, we need to focus on edge cases and scenarios that can maximize the failure probability. Here are the steps to consider:

1. **Test Case with Duplicates in the Template:**
   - Ensure the template array has duplicate values to test the one-to-one correspondence requirement.
2. **Test Case with Strings of the Same Length as the Template:**
   - Ensure the strings have the same length as the template array.
3. **Test Case with Strings that Do Not Match the Template:**
   - Ensure some strings do not match the template to test the "NO" output.
4. **Test Case with Strings that Match the Template:**
   - Ensure some strings match the template to test the "YES" output.
5. **Test Case with Edge Cases:**
   - Test with the minimum and maximum values for the template and strings.

Here is a small-scale test input that covers these scenarios:

```plaintext
3
5
3 5 2 1 3
2
abfda
afbfa
4
1 2 1 2
3
abab
abcd
aabb
2
1 1
2
aa
ab
```

Explanation:
1. **First Test Case:**
   - Template: [3, 5, 2, 1, 3]
   - Strings: ["abfda", "afbfa"]
   - "abfda" matches the template (YES).
   - "afbfa" does not match the template (NO).

2. **Second Test Case:**
   - Template: [1, 2, 1, 2]
   - Strings: ["abab", "abcd", "aabb"]
   - "abab" matches the template (YES).
   - "abcd" does not match the template (NO).
   - "aabb" does not match the template (NO).

3. **Third Test Case:**
   - Template: [1, 1]
   - Strings: ["aa", "ab"]
   - "aa" matches the template (YES).
   - "ab" does not match the template (NO).

This test input should help expose any issues in the buggy code, particularly with the one-to-one correspondence requirement.

3
5
3 5 2 1 3
2
abfda
afbfa
4
1 2 1 2
3
abab
abcd
aabb
2
1 1
2
aa
ab


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
void solve()
{
int n;cin>>n;
std::vector<int> a(n);
for (int i = 0; i<n; ++i) cin>>a[i];
int m;cin>>m;
vector<string> s(m);
for (int i = 0; i < m; ++i)
{
    cin>>s[i];
    int x=0;
if(n!=s[i].size()) {cout<<"No"<<endl;continue;}
map<char,int> mp;
mp.clear();
for (int j = 0; j <s[i].size(); ++j)
{

    if(mp[s[i][j]]==0)
    mp[s[i][j]]=a[j];
    else if(mp[s[i][j]]!=a[j])
    {x=1;cout<<"No"<<endl;break;}

}
if(x==0) cout<<"Yes"<<endl;
}

}

int main()
{
    int t;cin>>t;
    while(t--)
    solve();
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on the logic that checks the one-to-one correspondence between the characters of the string and the elements of the array. The bug in the code is that it only checks if the same characters in the string map to the same numbers in the array, but it does not check the reverse: if the same numbers in the array map to the same characters in the string.

To create a test input that exposes this bug, we can use the following approach:

1. **Array and String Length**: Ensure the array and string have the same length.
2. **One-to-One Correspondence**: Create a string where the same numbers in the array map to the same characters, but different numbers map to the same character, which should be considered invalid.

Here is a small-scale test input that meets these criteria:

```plaintext
1
5
1 2 3 4 1
2
abcda
abcde
```

### Explanation:
- **Test Case 1**:
  - **Array**: [1, 2, 3, 4, 1]
  - **String 1**: "abcda"
    - The first and last characters ('a') correspond to the same number (1), which is correct.
    - The second character ('b') corresponds to 2, which is correct.
    - The third character ('c') corresponds to 3, which is correct.
    - The fourth character ('d') corresponds to 4, which is correct.
    - However, the string "abcda" is invalid because different numbers (1 and 4) map to the same character ('a').
  - **String 2**: "abcde"
    - Each character in the string corresponds to a unique number in the array, which is correct.
    - This string should be valid.

The expected output for this test input should be:
```plaintext
NO
YES
```

This test input will expose the bug in the code because the code will incorrectly output "Yes" for the first string "abcda".


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
t=int(input())
for x in range(t):
  n=int(input())
  a=list(map(int,input().split()))
  m=int(input())
  for x in range(m):
    s=input()
    if len(s)!=n:
      print('NO')
    else:
      dict1={}
      flag='YES'
      for x in range(n):
        if s[x] not in dict1:
          dict1[s[x]]=a[x]
        else:
          if dict1[s[x]]!=a[x]:
            flag='NO'
            break
      print(flag)

**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 scenarios where the one-to-one correspondence between characters in the string and numbers in the array is not maintained correctly. The buggy code only checks if the same characters in the string map to the same numbers in the array, but it does not check the reverse condition (i.e., the same numbers in the array should map to the same characters in the string).

Let's construct a test case that will fail the buggy code:

1. **Test Case with a Template and Strings:**
   - **Template:** A template with repeated numbers.
   - **Strings:**
     - One string that matches the template.
     - One string that does not match the template because it violates the one-to-one correspondence in the reverse direction.

Here is the test input:

```plaintext
2
5
3 5 2 1 3
2
abfda
afbfa
4
1 1 1 1
3
aaaa
abbb
aabb
```

Explanation:
- **First Test Case:**
  - Template: [3, 5, 2, 1, 3]
  - String "abfda" matches the template.
  - String "afbfa" does not match the template because 'f' maps to both 1 and 5, violating the one-to-one correspondence.

- **Second Test Case:**
  - Template: [1, 1, 1, 1]
  - String "aaaa" matches the template.
  - String "abbb" does not match the template because '1' maps to both 'a' and 'b', violating the one-to-one correspondence.
  - String "aabb" does not match the template because '1' maps to both 'a' and 'b', violating the one-to-one correspondence.

This test input will expose the bug in the buggy code because it does not check the reverse condition (i.e., the same numbers in the array should map to the same characters in the string).

2
5
3 5 2 1 3
2
abfda
afbfa
4
1 1 1 1
3
aaaa
abbb
aabb


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
t = int(input())

for _ in range(t):

    n = int(input())

    a = list(map(int, input().split()))

    m = int(input())

    strings = []
    for _ in range(m):
        strings.append(input())
    unique_a = list(set(a))
    for s in strings:
        s2 = ''.join(set(s))
        if len(s2)==len(unique_a) and len(s)==len(a):
            print("YES")
        else:
            print("NO")

**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 where the faulty logic can be easily observed. The main issue with the buggy code is that it only checks the length of the unique elements in the array and the string, but it does not check the one-to-one correspondence between the characters and the elements.

Let's consider the following edge cases:
1. **Strings with the same length as the array but different characters for the same values in the array.**
2. **Strings with the same length as the array but the same characters for different values in the array.**
3. **Strings with the same length as the array but a different number of unique characters.**

Here is a small-scale test input that covers these cases:

```plaintext
3
5
1 2 3 4 5
3
abcde
aabbccdde
abcde
2
1 1
2
aa
ab
4
1 2 1 2
4
abab
aabb
abba
abab
```

Explanation:
1. **Test Case 1:**
   - Array: [1, 2, 3, 4, 5]
   - Strings: "abcde", "aabbccdde", "abcde"
   - "abcde" should match because it has a one-to-one correspondence.
   - "aabbccdde" should not match because it has more characters than the array length.
   - "abcde" should match because it has a one-to-one correspondence.

2. **Test Case 2:**
   - Array: [1, 1]
   - Strings: "aa", "ab"
   - "aa" should match because it has a one-to-one correspondence.
   - "ab" should not match because it does not have the same characters for the same values in the array.

3. **Test Case 3:**
   - Array: [1, 2, 1, 2]
   - Strings: "abab", "aabb", "abba", "abab"
   - "abab" should match because it has a one-to-one correspondence.
   - "aabb" should not match because it does not have the same characters for the same values in the array.
   - "abba" should not match because it does not have the same characters for the same values in the array.
   - "abab" should match because it has a one-to-one correspondence.

This test input should effectively expose the bug in the provided buggy code.

3
5
1 2 3 4 5
3
abcde
aabbccdde
abcde
2
1 1
2
aa
ab
4
1 2 1 2
4
abab
aabb
abba
abab


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Numeric String Template
time_limit_ms: 2000
memory_limit_mb: 256
Description: Kristina has an array $a$, called a template, consisting of $n$ integers. She also has $m$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $1$ to $m$. She wants to check which strings match the template.

A string $s$ is considered to match the template if all of the following conditions are simultaneously satisfied:

  * The length of the string $s$ is equal to the number of elements in the array $a$.   * The same numbers from $a$ correspond to the same symbols from $s$. So, if $a_i = a_j$, then $s_i = s_j$ for ($1 \le i, j \le n$).   * The same symbols from $s$ correspond to the same numbers from $a$. So, if $s_i = s_j$, then $a_i = a_j$ for ($1 \le i, j \le n$).

In other words, there must be a one-to-one correspondence between the characters of the string and the elements of the array.

For example, if $a$ = [$3, 5, 2, 1, 3$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $1$ and $5$.

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

The following descriptions are for the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($-10^9 \le a_i \le 10^9$) — the elements of the array $a$.

The third line of each test case contains a single integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of strings to check for template matching.

Following are $m$ strings, each containing a non-empty string $s_j$ ($1 \le |s_j| \le 2 \cdot 10^5$), consisting of lowercase Latin letters.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$, and that the sum of the lengths of all strings does not exceed $2 \cdot 10^5$.

Output: For each test case, output $m$ lines. On the $i$-th line ($1 \le i \le m$) output:

  * "YES", if the string with index $i$ matches the template;    * "NO" otherwise.

You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).

Examples:
input:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
output:
YES
NO
YES
NO
NO
NO
YES
NO
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
import java.io.*;
import java.util.HashMap;
import java.util.InputMismatchException;
import java.util.Map;

public class C {
    static FastIO io = new FastIO();
    static int mod = 1_000_000_007;

    public static void main(String[] args) {
        int tc = io.nextInt();
        while (tc-- > 0) {
            int n = io.nextInt();
            int[] arr = new int[n];
            for (int i = 0; i < n; i++) {
                arr[i] = io.nextInt();
            }
            int m = io.nextInt();
            for (int i = 0; i < m; i++) {
                String str = io.next();
                io.println(isMatching(n, arr, str) ? "YES" : "NO");
            }
        }
        io.close();
    }

    public static boolean isMatching(int n, int[] arr, String str) {
        if (str.length() != n) return false;
        Map<Character, Integer> map1 = new HashMap<>();
        Map<Integer, Character> map2 = new HashMap<>();
        for (int i = 0; i < n; i++) {
            char curChar = str.charAt(i);
            int curInt = arr[i];
            if (!map2.containsKey(curInt)) map2.put(curInt, curChar);
            if (!map1.containsKey(curChar)) {
                map1.put(curChar, arr[i]);
                continue;
            }
            if (map1.containsKey(curChar) && map1.get(curChar) != curInt || map2.containsKey(curInt) && map2.get(curInt) != curChar) return false;
        }
        return true;
    }
}

class FastIO extends PrintWriter {
    private final InputStream stream;
    private final byte[] buf = new byte[1 << 16];
    private int curChar, numChars;

    // standard input
    public FastIO() {
        this(System.in, System.out);
    }

    public FastIO(InputStream i, OutputStream o) {
        super(o);
        stream = i;
    }

    // file input
    public FastIO(String i, String o) throws IOException {
        super(new FileWriter(o));
        stream = new FileInputStream(i);
    }

    // throws InputMismatchException() if previously detected end of file
    private int nextByte() {
        if (numChars == -1) throw new InputMismatchException();
        if (curChar >= numChars) {
            curChar = 0;
            try {
                numChars = stream.read(buf);
            } catch (IOException e) {
                throw new InputMismatchException();
            }
            if (numChars == -1) return -1; // end of file
        }
        return buf[curChar++];
    }

    // to read in entire lines, replace c <= ' '
    // with a function that checks whether c is a line break
    public String next() {
        int c;
        do {
            c = nextByte();
        } while (c <= ' ');
        StringBuilder res = new StringBuilder();
        do {
            res.appendCodePoint(c);
            c = nextByte();
        } while (c > ' ');
        return res.toString();
    }

    public int nextInt() {
        int c;
        do {
            c = nextByte();
        } while (c <= ' ');
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = nextByte();
        }
        int res = 0;
        do {
            if (c < '0' || c > '9') throw new InputMismatchException();
            res = 10 * res + c - '0';
            c = nextByte();
        } while (c > ' ');
        return res * sgn;
    }

    public long nextLong() {
        int c;
        do {
            c = nextByte();
        } while (c <= ' ');
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = nextByte();
        }
        long res = 0;
        do {
            if (c < '0' || c > '9') throw new InputMismatchException();
            res = 10 * res + c - '0';
            c = nextByte();
        } while (c > ' ');
        return res * sgn;
    }

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

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that maximize the failure probability. The main issue in the buggy code is the handling of the one-to-one correspondence between the characters of the string and the elements of the array. We need to ensure that the test input will highlight this issue.

### Step-by-Step Analysis:
1. **Edge Case: Different Characters with the Same Value in the Array**
   - This case will test whether the code correctly handles situations where different characters in the string correspond to the same value in the array.
2. **Edge Case: Same Character with Different Values in the Array**
   - This case will test whether the code correctly handles situations where the same character in the string corresponds to different values in the array.
3. **Edge Case: Empty or Single-Element Array**
   - This case will test the code's ability to handle edge cases with minimal input.
4. **Edge Case: Large Input**
   - This case will test the code's performance and correctness with large inputs, ensuring it does not exceed time or memory limits.

### Challenging Test Input:
```plaintext
4
5
1 2 3 4 5
2
abcde
abcda
3
1 1 1
3
aaa
aab
aba
4
1 2 1 2
4
abab
acac
abac
abca
2
1 2
3
ab
ac
aa
```

### Explanation:
1. **Test Case 1:**
   - Array: [1, 2, 3, 4, 5]
   - Strings: "abcde", "abcda"
   - "abcde" should match because all characters are unique and correspond to unique values in the array.
   - "abcda" should not match because 'a' corresponds to both 1 and 5.

2. **Test Case 2:**
   - Array: [1, 1, 1]
   - Strings: "aaa", "aab", "aba"
   - "aaa" should match because all characters are the same and correspond to the same value in the array.
   - "aab" should not match because 'a' and 'b' correspond to the same value 1.
   - "aba" should not match because 'a' and 'b' correspond to the same value 1.

3. **Test Case 3:**
   - Array: [1, 2, 1, 2]
   - Strings: "abab", "acac", "abac", "abca"
   - "abab" should match because 'a' and 'b' correspond to 1 and 2 respectively.
   - "acac" should match because 'a' and 'c' correspond to 1 and 2 respectively.
   - "abac" should not match because 'a' corresponds to both 1 and 2.
   - "abca" should not match because 'a' corresponds to both 1 and 2.

4. **Test Case 4:**
   - Array: [1, 2]
   - Strings: "ab", "ac", "aa"
   - "ab" should match because 'a' and 'b' correspond to 1 and 2 respectively.
   - "ac" should match because 'a' and 'c' correspond to 1 and 2 respectively.
   - "aa" should not match because 'a' corresponds to both 1 and 2.

This test input is designed to cover a variety of edge cases and should help in identifying the bug in the provided code.

4
5
1 2 3 4 5
2
abcde
abcda
3
1 1 1
3
aaa
aab
aba
4
1 2 1 2
4
abab
acac
abac
abca
2
1 2
3
ab
ac
aa