**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Digital Village (Extreme Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: This is the extreme version of the problem. In the three versions, the constraints on $n$ and $m$ are different. You can make hacks only if all the versions of the problem are solved.

Pak Chanek is setting up internet connections for the village of Khuntien. The village can be represented as a connected simple graph with $n$ houses and $m$ internet cables connecting house $u_i$ and house $v_i$, each with a latency of $w_i$.

There are $p$ houses that require internet. Pak Chanek can install servers in at most $k$ of the houses. The houses that need internet will then be connected to one of the servers. However, since each cable has its latency, the latency experienced by house $s_i$ requiring internet will be the maximum latency of the cables between that house and the server it is connected to.

For each $k = 1,2,\ldots,n$, help Pak Chanek determine the minimum total latency that can be achieved for all the houses requiring internet.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains 3 integers $n$, $m$, $p$ ($2 \le n \le 2 \cdot 10^5$; $n-1 \le m \le 2 \cdot 10^5$; $1 \le p \le n$) — the number of houses, the number of cables, and the number of houses that need internet.

The second line of each test case contains $p$ integers $s_1, s_2, \ldots, s_p$ ($1 \le s_i \le n$) — the houses that need internet. It is guaranteed that all elements of $s$ are distinct.

The $i$-th of the next $m$ lines of each test case contains three integers $u_i$, $v_i$, and $w_i$ ($1 \le u_i < v_i \le n$; $1 \le w_i \le 10^9$) — the internet cable connecting house $u_i$ and house $v_i$ with latency of $w_i$. It is guaranteed that the given edges form a connected simple graph.

It is guaranteed that the sum of $n$ and the sum of $m$ do not exceed $2 \cdot 10^5$.

Output: For each test case, output $n$ integers: the minimum total latency that can be achieved for all the houses requiring internet for each $k = 1,2,\ldots,n$.

Examples:
input:
2
9 8 5
2 5 6 8 9
1 2 1
1 3 2
3 4 10
4 5 3
4 6 5
1 7 10
7 8 4
7 9 2
3 3 2
3 1
1 2 1
2 3 3
1 3 2
output:
34 19 9 4 0 0 0 0 0
2 0 0

Note: In the first test case for $k=3$, a possible optimal solution is to install servers at vertices $2$, $6$ and $8$ and obtain the following latency:

  * $\text{latency}(2) = 0$    * $\text{latency}(5) = \max(3, 5) = 5$    * $\text{latency}(6) = 0$    * $\text{latency}(8) = 0$    * $\text{latency}(9) = \max(2, 4) = 4$

So the total latency is $9$.

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

typedef long long ll;

// Structure to represent an edge
struct Edge {
    int u, v;
    int w;
    bool operator<(const Edge& other) const {
        return w < other.w;
    }
};

// Union-Find (Disjoint Set Union - DSU) Structure for Kruskal's Algorithm
struct DSU {
    vector<int> parent;
    DSU(int n){
        parent.resize(n+1);
        for(int i=0;i<=n;i++) parent[i] = i;
    }
    int find_set(int x){
        if(parent[x] != x)
            parent[x] = find_set(parent[x]);
        return parent[x];
    }
    void union_set(int x, int y){
        int fx = find_set(x);
        int fy = find_set(y);
        if(fx != fy){
            parent[fy] = fx;
        }
    }
};

int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int t;
    cin >> t;
    while(t--){
        int n, m, p;
        cin >> n >> m >> p;
        vector<int> S(p);
        for(auto &x:S) cin >> x;
        // Read all edges
        vector<Edge> edges(m);
        for(int i=0;i<m;i++) cin >> edges[i].u >> edges[i].v >> edges[i].w;
        // Sort edges for Kruskal's algorithm
        sort(edges.begin(), edges.end());
        // Initialize MST adjacency list
        vector<vector<pair<int, int>>> adjMST(n+1, vector<pair<int, int>>());
        // Build MST using Kruskal's algorithm
        DSU dsu(n);
        for(auto &e: edges){
            if(dsu.find_set(e.u) != dsu.find_set(e.v)){
                dsu.union_set(e.u, e.v);
                adjMST[e.u].emplace_back(make_pair(e.v, e.w));
                adjMST[e.v].emplace_back(make_pair(e.u, e.w));
            }
        }
        // Preprocess LCA and maxEdge using Binary Lifting
        // Define LOG based on n
        int LOG = 1;
        while((1 << LOG) <= n) LOG++;
        // Initialize tables
        vector<vector<int>> up(n+1, vector<int>(LOG, 0));
        vector<vector<int>> maxEdge(n+1, vector<int>(LOG, 0));
        vector<int> depth_node(n+1, 0);
        // DFS to populate up and maxEdge tables
        // Use a lambda for DFS
        auto dfs = [&](auto &dfs_ref, int u, int parent_node, int edge_weight) -> void {
            up[u][0] = parent_node;
            maxEdge[u][0] = edge_weight;
            for(int k=1; k<LOG; k++){
                up[u][k] = up[ up[u][k-1] ][k-1];
                maxEdge[u][k] = max(maxEdge[u][k-1], maxEdge[ up[u][k-1] ][k-1]);
            }
            for(auto &[v, w]: adjMST[u]){
                if(v != parent_node){
                    depth_node[v] = depth_node[u] +1;
                    dfs_ref(dfs_ref, v, u, w);
                }
            }
        };
        // Assume node 1 is the root
        dfs(dfs, 1, 1, 0);
        // Function to compute maximum edge on path from u to v
        auto get_max_edge = [&](int u, int v) -> int {
            if(u == v) return 0;
            int max_w = 0;
            if(depth_node[u] < depth_node[v]) swap(u, v);
            // Bring u to the same depth as v
            for(int k=LOG-1; k>=0; k--){
                if(depth_node[u] - (1<<k) >= depth_node[v]){
                    max_w = max(max_w, maxEdge[u][k]);
                    u = up[u][k];
                }
            }
            if(u == v) return max_w;
            // Now climb both u and v
            for(int k=LOG-1; k>=0; k--){
                if(up[u][k] != up[v][k]){
                    max_w = max({max_w, maxEdge[u][k], maxEdge[v][k]});
                    u = up[u][k];
                    v = up[v][k];
                }
            }
            // Finally, take the edge to LCA
            max_w = max({max_w, maxEdge[u][0], maxEdge[v][0]});
            return max_w;
        };
        // Compute d[s][c] for all required houses and all c
        // d is a vector of vectors, size p x (n+1)
        vector<vector<int>> d(p, vector<int>(n+1, 0));
        for(int i=0; i<p; i++){
            int s = S[i];
            for(int c=1; c<=n; c++){
                d[i][c] = get_max_edge(s, c);
            }
        }
        // Greedy Server Selection
        // Initialize min_d for each required house
        vector<int> min_d(p, INT32_MAX);
        // Initialize selected servers
        vector<bool> selected(n+1, false);
        // Initialize answer array
        vector<ll> answer(n, 0);
        // Iterate for k from 1 to p
        for(int k=1; k<=p; k++){
            // Find the best server to add
            int best_c = -1;
            ll best_sum = LLONG_MAX;
            for(int c=1; c<=n; c++){
                if(selected[c]) continue;
                ll temp_sum =0;
                for(int s=0; s<p; s++) {
                    temp_sum += min( (ll)min_d[s], (ll)d[s][c]);
                    if(temp_sum >= best_sum) break; // Early termination
                }
                if(temp_sum < best_sum){
                    best_sum = temp_sum;
                    best_c = c;
                }
            }
            // Select the best server
            selected[best_c] = true;
            // Update min_d
            for(int s=0; s<p; s++) {
                min_d[s] = min(min_d[s], d[s][best_c]);
            }
            // Compute the new sum
            ll current_sum =0;
            for(int s=0; s<p; s++) current_sum += (ll)min_d[s];
            // Assign to answer
            answer[k-1] = current_sum;
        }
        // For k = p+1 to n, sum is 0
        for(int k=p; k<n; k++) answer[k] = 0;
        // Output the answer
        for(int k=0; k<n; k++) cout << answer[k] << (k < n-1 ? ' ' : '\n');
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Digital Village (Extreme Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: This is the extreme version of the problem. In the three versions, the constraints on $n$ and $m$ are different. You can make hacks only if all the versions of the problem are solved.

Pak Chanek is setting up internet connections for the village of Khuntien. The village can be represented as a connected simple graph with $n$ houses and $m$ internet cables connecting house $u_i$ and house $v_i$, each with a latency of $w_i$.

There are $p$ houses that require internet. Pak Chanek can install servers in at most $k$ of the houses. The houses that need internet will then be connected to one of the servers. However, since each cable has its latency, the latency experienced by house $s_i$ requiring internet will be the maximum latency of the cables between that house and the server it is connected to.

For each $k = 1,2,\ldots,n$, help Pak Chanek determine the minimum total latency that can be achieved for all the houses requiring internet.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains 3 integers $n$, $m$, $p$ ($2 \le n \le 2 \cdot 10^5$; $n-1 \le m \le 2 \cdot 10^5$; $1 \le p \le n$) — the number of houses, the number of cables, and the number of houses that need internet.

The second line of each test case contains $p$ integers $s_1, s_2, \ldots, s_p$ ($1 \le s_i \le n$) — the houses that need internet. It is guaranteed that all elements of $s$ are distinct.

The $i$-th of the next $m$ lines of each test case contains three integers $u_i$, $v_i$, and $w_i$ ($1 \le u_i < v_i \le n$; $1 \le w_i \le 10^9$) — the internet cable connecting house $u_i$ and house $v_i$ with latency of $w_i$. It is guaranteed that the given edges form a connected simple graph.

It is guaranteed that the sum of $n$ and the sum of $m$ do not exceed $2 \cdot 10^5$.

Output: For each test case, output $n$ integers: the minimum total latency that can be achieved for all the houses requiring internet for each $k = 1,2,\ldots,n$.

Examples:
input:
2
9 8 5
2 5 6 8 9
1 2 1
1 3 2
3 4 10
4 5 3
4 6 5
1 7 10
7 8 4
7 9 2
3 3 2
3 1
1 2 1
2 3 3
1 3 2
output:
34 19 9 4 0 0 0 0 0
2 0 0

Note: In the first test case for $k=3$, a possible optimal solution is to install servers at vertices $2$, $6$ and $8$ and obtain the following latency:

  * $\text{latency}(2) = 0$    * $\text{latency}(5) = \max(3, 5) = 5$    * $\text{latency}(6) = 0$    * $\text{latency}(8) = 0$    * $\text{latency}(9) = \max(2, 4) = 4$

So the total latency is $9$.

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

using namespace std;

template <typename U, typename V, typename W>
struct Compare {
    bool operator()(const tuple<U, V, W>& a, const tuple<U, V, W>& b) const {
        return get<2>(a) > get<2>(b);
    }
};

template <typename U, typename V, typename W, typename Container = vector<tuple<U, V, W>>>
using PQ = priority_queue<tuple<U, V, W>, Container, Compare<U, V, W>>;

pair<int, int> findServer(vector<int>& servers, vector<int>& targets, int user) {
    if (servers[user] != user) {
        auto [v, w] = findServer(servers, targets, servers[user]);
        servers[user] = v;
        targets[user] = max(targets[user], w);
    }
    return { servers[user], targets[user] };
}

void insertQueue(
    int user,
    int server,
    int latency,
    PQ<int, int, long long>& paths,
    vector<pair<vector<pair<int, int>>, vector<pair<int, int>>>>& hosts,
    vector<long long>& acc,
    vector<int>& targets
) {
    vector<pair<int, int>> clients;
    long long new_acc = 0;
    int count = 0;
    for (auto& p : hosts[user].first) {
        if (p.first <= latency) {
            count += p.second;
        } else {
            clients.emplace_back(p);
            new_acc += (long long)p.first * p.second;
        }
    }
    clients.emplace_back(pair<int, int> { latency, count });
    paths.push({ user, server, new_acc + (long long)latency * count - acc[user] });
    hosts[user].second = move(clients);
    targets[user] = latency;
}

pair<int, int> findNearServer(
    int user,
    PQ<int, int, int>& pq,
    unordered_set<int>& mem,
    vector<vector<pair<int, int>>>& edges,
    vector<int>& servers,
    vector<int>& targets,
    vector<int>& dp
) {
    mem = unordered_set<int>();
    while (!pq.empty()) {
        auto [u, v, w] = pq.top();
        pq.pop();
        auto [sv, sw] = findServer(servers, targets, v);
        if (sv && user != sv) return { sv, max(w, sw) };
        if (dp[v] || mem.count(v)) continue;
        mem.insert(v);
        for (auto [nv, nw] : edges[v]) if (nv != u && nv != user) {
            pq.push({ v, nv, max(w, nw) });
        }
    }
    return { 0, 0 };
}

int pCases = 0, pHouses = 0, pCables = 0, pUsers = 0;
int nCases = 0, nHouses = 0, nCables = 0, nUsers = 0;

void minLatency(int n, vector<tuple<int, int, int>>& cables, vector<int>& users) {
    vector<vector<pair<int, int>>> edges(n + 1);
    for (auto [u, v, w] : cables) {
        edges[u].emplace_back(pair<int, int> { v, w });
        edges[v].emplace_back(pair<int, int> { u, w });
    }

    PQ<int, int, long long> paths;
    vector<pair<PQ<int, int, int>, unordered_set<int>>> status(n + 1);
    vector<int> servers(n + 1);
    vector<int> targets(n + 1);
    vector<int> dp(n + 1);
    vector<pair<vector<pair<int, int>>, vector<pair<int, int>>>> hosts(n + 1);
    vector<long long> acc(n + 1);
    for (int u : users) {
        servers[u] = u;
        hosts[u].first.emplace_back(pair<int, int> { 0, 1 });
    }
    for (int u = 1; u <= n; u++) {
        auto& pq = status[u].first;
        for (auto [v, w] : edges[u]) pq.push({ u, v, w });
        auto [v, w] = findNearServer(u, pq, status[u].second, edges, servers, targets, dp);
        if (servers[u] == u) {
            insertQueue(u, v, w, paths, hosts, acc, targets);
        } else {
            servers[u] = v;
            targets[u] = w;
            status[u] = pair<PQ<int, int, int>, unordered_set<int>>();
            hosts[u] = pair<vector<pair<int, int>>, vector<pair<int, int>>>();
        }
    }

    vector<bool> updated(n + 1);
    vector<long long> mins(n);
    for (int k = users.size() - 2; k >= 0; k--) {
        int u, v, sw;
        long long w;
        tie(u, v, w) = paths.top();
        paths.pop();
        tie(v, sw) = findServer(servers, targets, v);
        while (updated[u] || u == v) {
            if (u != v) {
                insertQueue(u, v, targets[u], paths, hosts, acc, targets);
            } else {
                tie(v, w) = findNearServer(u, status[u].first, status[u].second, edges, servers, targets, dp);
                insertQueue(u, v, w, paths, hosts, acc, targets);
            }
            updated[u] = false;
            tie(u, v, w) = paths.top();
            paths.pop();
            tie(v, sw) = findServer(servers, targets, v);
        }
        mins[k] = mins[k + 1] + w;

        servers[u] = v;
        auto& mem = status[u].second;
        for (int nu : mem) dp[nu] = v;

        auto& pqu = status[u].first;
        auto& pqv = status[v].first;
        while (!pqu.empty()) {
            auto [nu, nv, nw] = pqu.top();
            pqu.pop();
            if (mem.count(nv)) continue;
            mem.insert(nv);
            pqv.push({ nu, nv, servers[nu] == v ? max(nw, targets[nu]) : max(nw, max(targets[nu], targets[u])) });
        }

        hosts[v].first.insert(hosts[v].first.end(), hosts[u].second.begin(), hosts[u].second.end());
        acc[v] += w + acc[u];
        updated[v] = true;

        status[u].first = PQ<int, int, int>();
        status[u].second = unordered_set<int>();
        hosts[u].first = vector<pair<int, int>>();
        hosts[u].second = vector<pair<int, int>>();
    }

    if (nCases == 20 && nHouses == 4519 && nCables == 4518 && nUsers == 2) {
        for (int i = cables.size() - 1; i >= 0; i--) {
            auto [u, v, w] = cables[i];
            if (u == 2131 && v == 3383 || u == 3383 && v == 2131) cout << u << " " << v << " ";
        }
        cout << endl;
        return;
    }

    for (long long i : mins) cout << i << " ";
    cout << endl;
}

int main() {
    int cases;
    cin >> cases;
    for (int i = 0; i < cases; i++) {
        int n, m, p;
        cin >> n >> m >> p;
        pCases = nCases, pHouses = nHouses, pCables = nCables, pUsers = nUsers;
        nCases = cases, nHouses = n, nCables = m, nUsers = p;
        vector<int> users(p);
        while (p-- > 0) cin >> users[p];
        vector<tuple<int, int, int>> edges;
        int u, v, w;
        while (m-- > 0) {
            cin >> u >> v >> w;
            edges.emplace_back(move(tuple<int, int, int> { u, v, w }));
        }
        minLatency(n, edges, users);
    }
    return 0;
}

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

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

Think step by step.

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

1. **Edge Cases**: Consider scenarios where the number of houses that need internet is very small or very large compared to the total number of houses.
2. **Graph Structure**: Create a graph that is not a simple tree but has multiple paths and cycles, which can make the server placement and latency calculation more complex.
3. **Latency Values**: Use latency values that are not uniformly distributed to test the algorithm's handling of extreme latency differences.

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

```plaintext
2
5 4 3
1 3 5
1 2 1
2 3 2
3 4 3
4 5 4
6 5 2
1 4
1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
```

### Explanation:
1. **Test Case 1**:
   - **5 houses, 4 cables, 3 houses needing internet**: This is a small graph with a linear structure.
   - **Houses needing internet**: 1, 3, 5.
   - **Cables and latencies**:
     - 1-2 with latency 1
     - 2-3 with latency 2
     - 3-4 with latency 3
     - 4-5 with latency 4

2. **Test Case 2**:
   - **6 houses, 5 cables, 2 houses needing internet**: This is a slightly larger graph with a linear structure.
   - **Houses needing internet**: 1, 4.
   - **Cables and latencies**:
     - 1-2 with latency 1
     - 2-3 with latency 2
     - 3-4 with latency 3
     - 4-5 with latency 4
     - 5-6 with latency 5

This test input is designed to challenge the algorithm by:
- Testing with a small number of houses and cables.
- Testing with a linear graph structure.
- Testing with a small number of houses needing internet.
- Using different latency values to ensure the algorithm handles varying latencies correctly.

This should help in identifying any issues 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: Digital Village (Extreme Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: This is the extreme version of the problem. In the three versions, the constraints on $n$ and $m$ are different. You can make hacks only if all the versions of the problem are solved.

Pak Chanek is setting up internet connections for the village of Khuntien. The village can be represented as a connected simple graph with $n$ houses and $m$ internet cables connecting house $u_i$ and house $v_i$, each with a latency of $w_i$.

There are $p$ houses that require internet. Pak Chanek can install servers in at most $k$ of the houses. The houses that need internet will then be connected to one of the servers. However, since each cable has its latency, the latency experienced by house $s_i$ requiring internet will be the maximum latency of the cables between that house and the server it is connected to.

For each $k = 1,2,\ldots,n$, help Pak Chanek determine the minimum total latency that can be achieved for all the houses requiring internet.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains 3 integers $n$, $m$, $p$ ($2 \le n \le 2 \cdot 10^5$; $n-1 \le m \le 2 \cdot 10^5$; $1 \le p \le n$) — the number of houses, the number of cables, and the number of houses that need internet.

The second line of each test case contains $p$ integers $s_1, s_2, \ldots, s_p$ ($1 \le s_i \le n$) — the houses that need internet. It is guaranteed that all elements of $s$ are distinct.

The $i$-th of the next $m$ lines of each test case contains three integers $u_i$, $v_i$, and $w_i$ ($1 \le u_i < v_i \le n$; $1 \le w_i \le 10^9$) — the internet cable connecting house $u_i$ and house $v_i$ with latency of $w_i$. It is guaranteed that the given edges form a connected simple graph.

It is guaranteed that the sum of $n$ and the sum of $m$ do not exceed $2 \cdot 10^5$.

Output: For each test case, output $n$ integers: the minimum total latency that can be achieved for all the houses requiring internet for each $k = 1,2,\ldots,n$.

Examples:
input:
2
9 8 5
2 5 6 8 9
1 2 1
1 3 2
3 4 10
4 5 3
4 6 5
1 7 10
7 8 4
7 9 2
3 3 2
3 1
1 2 1
2 3 3
1 3 2
output:
34 19 9 4 0 0 0 0 0
2 0 0

Note: In the first test case for $k=3$, a possible optimal solution is to install servers at vertices $2$, $6$ and $8$ and obtain the following latency:

  * $\text{latency}(2) = 0$    * $\text{latency}(5) = \max(3, 5) = 5$    * $\text{latency}(6) = 0$    * $\text{latency}(8) = 0$    * $\text{latency}(9) = \max(2, 4) = 4$

So the total latency is $9$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
using pii = pair<int,int>;
#define fi first
#define se second
const int N = 200000 + 200;
const long long inf = 0x3f3f3f3f3f3f3f3f;
long long tmp[2 * N];
void solve(){
    int n,m,p;
    cin>>n>>m>>p;
    vector<vector<int>>adj(2 * n + 1);
    vector<int>ned(2 * n + 1);
    vector<vector<long long>>f(2 * n + 1);
    vector<long long>val(2 * n + 1);
    vector<int>fa(2 * n + 1),siz(2 * n + 1),sz(2 * n + 1);
    for(int i = 0;i <= 2 * n;i ++)
        f[i].clear();
    iota(fa.begin(),fa.end(),0);
    for(int i = 1;i <= p;i ++){
        int x;
        cin>>x;
        ned[x] = 1;
    }
    int tot = n;
    vector<array<int,3>>edge;
    for(int i = 1;i <= m;i ++){
        int u,v,w;
        cin>>u>>v>>w;
        edge.push_back({w,u,v});
    }
    sort(edge.begin(),edge.end());
    for(int i = 1;i <= n;i ++)
        sz[i] = 1;
    auto find = [&](auto &&find,int x)->int
    {
        return fa[x] == x?x:fa[x] = find(find,fa[x]);
    };
    for(int i = 0;i < m;i ++){
        auto [w,u,v] = edge[i];
        int fu = find(find,u),fv = find(find,v);
        if(fu != fv){
            tot ++;
            fa[fu] = tot;
            fa[fv] = tot;
            sz[tot] = sz[fu] + sz[fv];
            if(sz[fu] < sz[fv]){
                adj[tot].push_back(fu);
                adj[tot].push_back(fv);
            } else {
                adj[tot].push_back(fv);
                adj[tot].push_back(fu);
            }
            val[tot] = w;
        }
    }
    auto dfs = [&](auto &&dfs,int x)->void
    {
        if(ned[x]){
            siz[x] = 1;
            f[x].push_back(inf);
            f[x].push_back(0);
        }
        bool ok = 0;
        for(auto v:adj[x]){
            dfs(dfs,v);
            if(siz[v] == 0)
                continue;
            if(!ok)
            {
                siz[x] = siz[v];
                f[x].swap(f[v]);
                ok = 1;
                continue;
            }
            for(int i = 0;i <= siz[x] + siz[v];i ++)
                tmp[i] = inf;
            for(int i = 1;i <= siz[v];i ++)
                tmp[i] = min(tmp[i],siz[x] * val[x] + f[v][i]);
            for(int i = 1;i <= siz[x];i ++)
                tmp[i] = min(tmp[i],siz[v] * val[x] + f[x][i]);
            int li = 1,ri = 1;
            for(int i = 2;i <= siz[x] + siz[v];i ++){
                tmp[i] = min(f[x][li] + f[v][ri],tmp[i]);
                if(li + 1 <= siz[x] && ri + 1 <= siz[v]){
                    if(f[x][li] - f[x][li + 1] >= f[v][ri] - f[v][ri + 1])
                        li ++;
                    else
                        ri ++;
                } else if(li + 1 <= siz[x])
                    li ++;
                else
                    ri ++;
            }
            f[x].resize(siz[x] + siz[v] + 1);
            for(int i = 0;i <= siz[x] + siz[v];i ++)
                f[x][i] = tmp[i];
            vector<long long>().swap(f[v]);
            siz[x] += siz[v];
        }
    };
    dfs(dfs,tot);
    f[tot][0] = inf;
    for(int i = 1;i <= n;i ++){
        if(i >= f[tot].size())
            cout<<0<<" ";
        else{
            f[tot][i] = min(f[tot][i],f[tot][i - 1]);
            cout<<f[tot][i]<<" ";
        }
    }
    cout<<"\n";
}
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    int T;
    cin>>T;
    while(T --)
        solve();
}

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

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

Think step by step.

Response:
```plaintext
1
4 4 3
1 3 4
1 2 1
2 3 1
3 4 1
1 4 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

void fun1(){
      int a, bb, ans=0; cin>>a>>bb;
      string s; cin>>s;
      vector<int>v(7,0);
      for(auto &b: s){
        v[b-'A']++;
      }
      for(auto &b: v){
          if(b<bb){ans+=bb-b;}
      }
      cout<<ans<<"\n";
}
void fun2(){
    int a,b,c; cin>>a>>b>>c;
    vector<int>v(a);
    for(auto &bb: v){cin>>bb;}
    int fv=v[b-1];
    v.push_back(0); v.push_back(101);
    sort(v.begin(), v.end(), greater<>());
    if(fv>v[c+1]){cout<<"YES\n"; return;}
    if(fv==v[c+1] and fv==v[c]){cout<<"MAYBE\n"; return;}
    cout<<"NO\n"; return;
    // 101 5 4 3 2 1 0
    // 3 2
}

void fun3(){
    string a, b; cin>>a>>b;
    swap(a[0],b[0]);
    cout<<a<<" "<<b<<"\n";
}
void fun4(){
    int n; cin>>n;
    if(n<=3){cout<<n<<"\n";}
    else{cout<<"2\n";}
}
void fun5(){
    long long n, ans=0; cin>>n;
    vector<long long>v(n);
    for(auto &b: v){cin>>b;}
    long long sum=0, mx=v[0];
    for(int i=0; i<v.size(); i++){
        if(v[i]==0){ans++; continue;}
        mx=max(mx, v[i]);
        sum+=v[i];
        long long p=sum-mx;
        if(p==mx){ans++;}
    }
    cout<<ans<<"\n";
}

int main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);

    int t; cin>>t;
    while(t--){
    //   fun1()
        // fun2();
        // fun3();
        // fun4();
        fun5();
    }

    return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the given code, we need to focus on edge cases and scenarios that are likely to cause the algorithm to fail. The key points to consider are:

1. **Zero Elements**: The code should handle cases where the array contains a single element that is zero.
2. **Multiple Zeros**: The code should handle cases where the array contains multiple zeros.
3. **Large Numbers**: The code should handle cases where the array contains very large numbers.
4. **Negative Sums**: The code should handle cases where the sum of elements can be negative (though the problem constraints state that elements are non-negative).
5. **Prefix Sums**: The code should correctly identify when a prefix sum matches one of its elements.

Given these considerations, a small-scale test input that can expose the bug is:

```plaintext
3
1
0
4
0 0 0 0
7
1 1 0 3 5 2 12
```

### Explanation:
1. **Test Case 1**:
   - Input: `1 0`
   - Expected Output: `1`
   - This tests the case where the array has a single element that is zero, which should be considered a good prefix.

2. **Test Case 2**:
   - Input: `4 0 0 0 0`
   - Expected Output: `4`
   - This tests the case where the array contains multiple zeros, and each prefix should be considered good.

3. **Test Case 3**:
   - Input: `7 1 1 0 3 5 2 12`
   - Expected Output: `4`
   - This tests the case where the array contains a mix of elements, and the code should correctly identify the good prefixes.

This test input should help in identifying any issues in the given code, especially in handling zeros and multiple good prefixes.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

int main(){
	ll test;
	cin>>test;
	while(test--){
		ll n;
        cin>>n;
        ll a[n+1];
        for(ll i=1;i<=n;i++) cin>>a[i];
        a[0]=0;
        ll max=0,c=0,sum=0;
        for(ll i=1;i<=n;i++){
            if(a[i]==0){ c++; continue;}
            if(a[i]>max){
                sum+=max;
                max=a[i];
            }
            else sum+=a[i];
            if(sum==max) c++;
        }
        cout<<c<<endl;
	}
}

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

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

Think step by step.

Response:
```plaintext
3
1
1
4
1 1 1 1
5
1 2 3 6 0
```

### Explanation:
1. **Test Case 1:**
   - **Input:** `1 1`
   - **Expected Output:** `0`
   - **Reason:** The prefix `[1]` is not good because the only element `1` cannot be represented as the sum of other elements (which are none).

2. **Test Case 2:**
   - **Input:** `4 1 1 1 1`
   - **Expected Output:** `0`
   - **Reason:** None of the prefixes `[1]`, `[1, 1]`, `[1, 1, 1]`, or `[1, 1, 1, 1]` are good. Each prefix sum is not equal to any of its elements.

3. **Test Case 3:**
   - **Input:** `5 1 2 3 6 0`
   - **Expected Output:** `3`
   - **Reason:**
     - The prefix `[1]` is not good.
     - The prefix `[1, 2]` is not good.
     - The prefix `[1, 2, 3]` is not good.
     - The prefix `[1, 2, 3, 6]` is good because `1 + 2 + 3 = 6`.
     - The prefix `[1, 2, 3, 6, 0]` is good because `1 + 2 + 3 + 6 = 12` and `0` is included, but the prefix `[1, 2, 3, 6]` is already good.

This test input is designed to expose the bug in the given buggy code by providing edge cases where the sum of elements is not straightforward and includes a zero element, which is a common source of errors in such problems.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
/*?????? ?? ???? ?? ???? ??? ????*/

#include<iostream>
#include<vector>
#include<string>
#include<cmath>
#include<algorithm>
#include<climits>
#include<stack>
#include<map>
#include<set>
#include<queue>

#define IOS ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0)
#define ll long long
#define sf(n) scanf("%d", &n)
#define sff(n, m) scanf("%d%d",&n,&m)

#define pf(n) printf("%d",n)
#define pff(n, m) printf("%d %d",n,m)
#define pp pair<int,int>

using namespace std;
ll gcd(ll a, ll b) { return b == 0 ? a : gcd(b, a % b); }
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; };
//#include <ext/pb_ds/assoc_container.hpp>
//#include <ext/pb_ds/tree_policy.hpp>
//using namespace __gnu_pbds;
//template<class T> using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
bool com(pair<ll, ll>a, pair<ll, ll>b) {
    if (abs(a.first - a.second) == abs(b.first - b.second)) {
        return a.first > b.first;
    }
    return abs(a.first - a.second) < abs(b.second - b.first);
}

int main()
{
    IOS;
    int test; cin >> test;
    while (test--) {
        int num; cin >> num;
        vector<int>v(num);
        set<int>st;
        for (int i = 0; i < num; i++)cin >> v[i];
        long long sum = 0,result=0;
        for (int i = 0; i < num; i++) {
            sum += v[i];

            long long k = sum / 2;

            if (sum % 2 == 0 && find(v.begin(), v.end(), k) != v.end()) {
                result++;

            }

        }

        cout << result << "\n";
    }
    return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases that can highlight the issues with the logic used in the algorithm. The main issue in the buggy code is that it incorrectly checks if the sum of the elements is even and then looks for a specific element in the array. This approach fails to correctly identify all good prefixes.

Here are the steps to generate a test input:

1. **Single Element Arrays**:
   - Arrays with a single element should be tested, especially with `0` and a non-zero element.

2. **Arrays with All Elements Equal**:
   - Arrays where all elements are the same can help test the sum and element checks.

3. **Arrays with a Large Number of Elements**:
   - Test with large arrays to ensure the code handles large inputs correctly.

4. **Arrays with Specific Patterns**:
   - Arrays where the sum of some elements equals another element can help test the logic for identifying good prefixes.

5. **Arrays with Zeros**:
   - Arrays with zeros can test edge cases where the sum is zero or where zeros are part of the sum.

Based on these considerations, a small-scale test input that can expose the bug is:

```plaintext
4
1
0
1
1
3
1 1 2
5
0 1 2 1 4
```

This test input includes:
- A single element array with `0` (should be good).
- A single element array with `1` (should not be good).
- An array with three elements where the sum of two elements equals the third (should be good).
- An array with five elements, including a zero, where the sum of some elements equals another element (should be good).

This input should help identify the issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
for i in range(int(input())):
  n=int(input())
  l=list(map(int,input().split()))
  c=0
  s=l[0]
  f=False
  for j in range(n):
    p=l[:j+1]
    m=max(p)
    if f and p[-1]==0 and j!=n-1:
      c+=1
      f=False if l[j+1]!=0 else True
    elif sum(p)==2*m:
      c+=1
      f=True
  if l!=[0]:
    print(c)
  else:
    print(1)

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

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

Think step by step.

Response:
```plaintext
1
5
0 0 1 1 2
```

**Explanation:**
This test case is designed to challenge the buggy code by including multiple zeros and repeated values. The prefixes of this array are:
- Prefix [0] is good.
- Prefix [0, 0] is good.
- Prefix [0, 0, 1] is not good.
- Prefix [0, 0, 1, 1] is good.
- Prefix [0, 0, 1, 1, 2] is good.

Thus, the correct output should be 4. The buggy code may fail to correctly handle the multiple zeros and repeated values, leading to an incorrect count.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
t=int(input())
for a in range(t):
    n=int(input())
    e=input()
    x=e.split()
    sum1=0
    max1=0
    good=0
    for i in range(n):
        sum1+=int(x[i])
        if int(x[i])>=max1:
            max1=i
        if int(x[max1])  == sum1-int(x[max1]):
            good+=1
    print(good)

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

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

Think step by step.

Response:
```plaintext
5
1
0
2
1 1
4
1 2 1 4
7
1 2 3 1 2 3 6
10
0 1 2 3 4 5 6 7 8 36
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

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

    sums = [sum(a[0:i+1]) for i in range(n)]
    #print(sums)

    numGoodArrays = 0
    for i in range(n):
        if sums[i]%2 == 0:
            if sums[i]/2 in a:
                numGoodArrays += 1
    print(f'{numGoodArrays}')

**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
6
1 1 2 4 8 16
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

public class GoodPrefixes {
    public static void main(String[] args){
        try(BufferedReader br = new BufferedReader(new InputStreamReader(System.in))){
            int t = Integer.parseInt(br.readLine());

            for(int i=0; i<t; i++){
                int n = Integer.parseInt(br.readLine());
                int[] inputArr = Arrays.stream(br.readLine().trim().split("\\s+")).mapToInt(Integer::parseInt).toArray();
                System.out.println(solve(n, inputArr));
            }
        }
        catch(IOException ex){
            throw new RuntimeException(ex);
        }
    }

    static int solve(int n, int[] input){

        int res = 0;
        long[] prefixSums = new long[n];
        boolean[] isGood = new boolean[n];

        prefixSums[0] = input[0];

        for(int i=1; i<n; i++){
            prefixSums[i] = prefixSums[i-1] + input[i];
        }

        if(input[0] == 0){
            res++;
            isGood[0] = true;
        }

        for(int i=1; i<n; i++){

            if(isGood[i-1]){
                if(input[i] == prefixSums[i-1])
                    isGood[i] = true;
            }else {
                for (int j = 0; j <= i; j++) {
                    long lsum = 0;
                    long rsum = 0;

                    if (j - 1 >= 0) {
                        lsum += prefixSums[j - 1];
                    }

                    rsum += prefixSums[i] - prefixSums[j];

                    if (lsum + rsum == input[j]) {
                        res++;
                        break;
                    }
                }
            }
        }

        return res;
    }
}

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

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

Think step by step.

Response:
```plaintext
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1 1 1 1 1 1 6
10
0 1 1 2 3 5 8 13 21 34
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
import java.util.*;
import java.io.*;
public class GoodPrefixes
{
    public static void main(String[] args) throws IOException
    {
      BufferedReader br=new BufferedReader(new InputStreamReader(System.in));
  		PrintWriter pw=new PrintWriter(System.out);
      int t=Integer.parseInt(br.readLine());
      for(int i=0;i<t;i++)
      {
        int n=Integer.parseInt(br.readLine());
        int[]arr=new int[n];
        long[]prefix=new long[n];
        StringTokenizer st=new StringTokenizer(br.readLine());
        for(int k=0;k<n;k++)
        arr[k]=Integer.parseInt(st.nextToken());
        prefix[0]=arr[0];
        for(int k=1;k<n;k++)
        prefix[k]=arr[k]+prefix[k-1];
        int max=arr[0];
        int maxindex=0;
        int res=0;
        for(int k=0;k<n;k++)
        {
          if(arr[k]>=max)
          {
            max=arr[k];
            maxindex=k;
          }
          if(maxindex==0)
          {
            if(prefix[maxindex]+prefix[k]==max)
            res++;
          }
          else if(prefix[maxindex-1]+prefix[k]-prefix[maxindex]==max)
          {
            res++;
          }
        }
        pw.println(res);
      }
      pw.close();
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's consider the following points:

1. **Edge Cases**:
   - Arrays with a single element.
   - Arrays with all elements being zero.
   - Arrays where the sum of all elements is zero.
   - Arrays where the sum of all elements is a single element.

2. **Specific Bug Analysis**:
   - The buggy code seems to have issues with how it tracks the maximum element and its index. It also has a flawed logic for checking if a prefix is good.

3. **Test Input Construction**:
   - We need to create a test input that will make the buggy code's logic fail. Specifically, we need to create a scenario where the maximum element and its index are not correctly identified, or where the sum of the prefix does not match the expected conditions.

Let's construct a test input that includes a mix of these edge cases and specific scenarios to expose the bug:

```plaintext
3
5
1 2 3 6 10
5
1 1 1 1 4
5
0 0 0 0 0
```

### Explanation:
1. **Test Case 1:**
   - Array: `[1, 2, 3, 6, 10]`
   - The prefix `[1, 2, 3, 6]` is good because `1 + 2 + 3 = 6`.
   - The prefix `[1, 2, 3, 6, 10]` is not good.
   - Expected output: `1`

2. **Test Case 2:**
   - Array: `[1, 1, 1, 1, 4]`
   - The prefix `[1, 1, 1, 1]` is not good.
   - The prefix `[1, 1, 1, 1, 4]` is good because `1 + 1 + 1 + 1 = 4`.
   - Expected output: `1`

3. **Test Case 3:**
   - Array: `[0, 0, 0, 0, 0]`
   - Every prefix is good because the sum of all elements in any prefix is zero.
   - Expected output: `5`

These test cases are designed to challenge the buggy code's logic for identifying good prefixes and handling edge cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
#include <iostream>
#include <vector>
using namespace std;

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int t;
    cin >> t;

    while (t--) {
        int n;
        cin >> n;
        vector<int> a(n);
        for (int i = 0; i < n; i++) {
            cin >> a[i];
        }

        int gp = 0;
        long long pp = 0;

        for (int i = 0; i < n; i++) {
            pp += a[i];
            for (int j = 0; j <= i; j++) {
                if (a[j] == pp - a[j]) {
                    gp++;
                    break;
                }
            }
        }

        cout << gp << endl;
    }

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
2
5
1 2 3 6 12
3
0 0 0
```

**Explanation:**

1. **Test Case 1:**
   - Array: `[1, 2, 3, 6, 12]`
   - This array is designed to test the handling of a sequence where the sum of the first few elements equals one of the elements. Specifically:
     - The prefix `[1, 2, 3]` is good because `1 + 2 + 3 = 6`.
     - The prefix `[1, 2, 3, 6]` is good because `1 + 2 + 3 + 6 = 12`.
     - The prefix `[1, 2, 3, 6, 12]` is good because `1 + 2 + 3 + 6 + 12 = 24` (but 12 is the last element and the sum of the others is 12).
   - The expected output is `3`.

2. **Test Case 2:**
   - Array: `[0, 0, 0]`
   - This array is designed to test the handling of multiple zeros, which should all be considered good prefixes because the sum of any prefix is always 0, and each 0 can be considered as the sum of the other elements (which is 0).
   - The expected output is `3`.

These test cases are small and cover edge cases where the sum of elements equals one of the elements and where multiple zeros are present, which should help in exposing any bugs in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int main()
{
    long long t;
    cin >> t;

    while (t--)
    {
        long long n;
        cin >> n;
        vector<long long> v(n);
        for (long long i = 0; i < n; i++)
        {
            cin >> v[i];
        }

        long long ans =0;
        long long maxi =0;
        if(v[0]==0){
            ans=1;
        }
        long long sum;
        for(long long i=1;i<n;i++){
            sum =0;
            auto m = max_element(v.begin(),v.begin()+i+1);
            maxi =*m;
            for(long long j=0;j<=i;j++){
                sum+=v[j];

            }
            sum -=maxi;
            if(sum==maxi){
                ans++;
            }
        }
        cout<<ans<<endl;

    }

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
2
5
1 2 3 6 0
4
0 0 0 0
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
def main():
  t = int(input().strip())
  for _ in range(t):
      n = int(input().strip())
      a = list(map(int, input().strip().split()))
      #b = list(map(int, input().strip().split()))
      #a, b = input().strip().split()
      count = 0
      for i in range(n):
          ar = a[0:i+1]
          m = max(ar)
          #arm = argmax(ar)
          ar.remove(m)
          if (sum(ar)==m):
              count += 1
      print(count)

if __name__ == "__main__":
  main()

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

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

Think step by step.

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

1. **Edge Cases:**
   - Arrays with a single element.
   - Arrays where all elements are the same.
   - Arrays where the sum of all elements is zero.
   - Arrays with multiple good prefixes.

2. **Scenarios that Maximize Failure Probability:**
   - Arrays where the maximum element is not unique and appears multiple times.
   - Arrays where the sum of the remaining elements equals the maximum element in multiple prefixes.

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

```plaintext
5
1
0
3
1 1 2
4
1 1 1 3
5
0 1 2 1 4
6
1 1 0 2 2 4
```

### Explanation:
1. **Test Case 1:**
   - Input: `1 0`
   - Expected Output: `1`
   - Reason: The array `[0]` is a good prefix.

2. **Test Case 2:**
   - Input: `3 1 1 2`
   - Expected Output: `2`
   - Reason: The prefixes `[1]` and `[1, 1, 2]` are good.

3. **Test Case 3:**
   - Input: `4 1 1 1 3`
   - Expected Output: `3`
   - Reason: The prefixes `[1]`, `[1, 1]`, and `[1, 1, 1, 3]` are good.

4. **Test Case 4:**
   - Input: `5 0 1 2 1 4`
   - Expected Output: `3`
   - Reason: The prefixes `[0]`, `[0, 1, 2, 1]`, and `[0, 1, 2, 1, 4]` are good.

5. **Test Case 5:**
   - Input: `6 1 1 0 2 2 4`
   - Expected Output: `4`
   - Reason: The prefixes `[1]`, `[1, 1, 0, 2, 2]`, and `[1, 1, 0, 2, 2, 4]` are good.

This test input should effectively expose the bug in the buggy code by including various edge cases and scenarios that maximize the failure probability.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
import sys
def input(): return sys.stdin.readline().strip()

t = int(input())
for _ in range(t):
    n = int(input())
    arr = list(map(int, input().split()))

    prefSum = [0]*(n+1)
    for i in range(1, n+1):
        prefSum[i] = prefSum[i-1]+arr[i-1]
    count = 0
    for i in range(1, n+1):
        for j in range(i):
            if arr[j] == prefSum[i]-arr[j]:
                count += 1
                break

    print(count)

**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 think through this:

1. **Edge Cases:**
   - Arrays with a single element.
   - Arrays with all elements being zero.
   - Arrays with all elements being the same non-zero value.
   - Arrays with a mix of zeros and non-zeros.
   - Arrays where the sum of elements is zero.

2. **Scenarios to Maximize Failure:**
   - Ensure the prefix sums are correctly calculated.
   - Test cases where the sum of elements in the prefix can be represented by a single element in the prefix.
   - Test cases where the sum of elements in the prefix cannot be represented by any single element.

3. **Small-Scale Test Input:**
   - We need a test input that is small enough to be valid and complete but still challenging enough to expose the bug.

Given these considerations, a good test input would be an array where the sum of elements in the prefix can be represented by a single element in the prefix, and where the prefix sums are not straightforward.

Here is a small-scale test input that should expose the bug:

```plaintext
1
6
1 2 3 6 0 12
```

Explanation:
- The array is `[1, 2, 3, 6, 0, 12]`.
- The prefixes are:
  - `[1]` (not good)
  - `[1, 2]` (not good)
  - `[1, 2, 3]` (good, since `6 = 1 + 2 + 3`)
  - `[1, 2, 3, 6]` (good, since `6 = 1 + 2 + 3`)
  - `[1, 2, 3, 6, 0]` (good, since `6 = 1 + 2 + 3 + 0`)
  - `[1, 2, 3, 6, 0, 12]` (good, since `12 = 1 + 2 + 3 + 6 + 0`)

This test input should reveal the bug in the code where the prefix sums are not correctly checked for the condition of being a good prefix.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
import java.util.Scanner;
public class Main
{
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int t = sc.nextInt();
		while(t-- >0){
		    int n = sc.nextInt();
		    long[] arr = new long[n];
		    for(int i=0;i<n;i++){
		        arr[i] = sc.nextLong();
		    }

		    long prefix = 0;
		    int count = 0;

		    for(int i=0;i<n;i++){
		        prefix += arr[i];

		        boolean isGood = false;
		        for(int j=0;j<=i;j++){
		            long sum = prefix - arr[j];
		            if(arr[j] == sum){
		                isGood = true;
		                break;
		            }
		        }

		        if(isGood) count++;
		    }

		    System.out.println(count);
		}
	}
}

**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 maximize the failure probability. Here are the steps to consider:

1. **Edge Case: Single Element Array**:
   - Test with an array containing a single element, specifically `0` and `1`, to ensure the code handles these edge cases correctly.

2. **Edge Case: All Elements Zero**:
   - Test with an array where all elements are `0`. This should result in all prefixes being good.

3. **Edge Case: Large Sum**:
   - Test with an array where the sum of elements is very large, to ensure the code handles large numbers correctly.

4. **Edge Case: Repeated Elements**:
   - Test with an array where elements are repeated, to ensure the code handles repeated elements correctly.

5. **Edge Case: Mixed Elements**:
   - Test with an array containing a mix of elements, including `0`, to ensure the code handles mixed elements correctly.

6. **Edge Case: Long Array**:
   - Test with a long array to ensure the code handles large inputs efficiently.

Given the constraints and the need for a small-scale test input, let's create a test case that combines several of these edge cases:

```plaintext
3
1
0
1
1
7
1 1 0 3 5 2 12
```

Explanation:
- The first test case is a single element array with `0`, which should be good.
- The second test case is a single element array with `1`, which should not be good.
- The third test case is an array with a mix of elements, including `0`, which should result in 4 good prefixes.

This test input should help expose any issues in the buggy code, especially in handling single elements, mixed elements, and the presence of `0`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

public class Main {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        int t = Integer.parseInt(br.readLine());
        StringBuilder sb = new StringBuilder();

        while (t-- > 0) {
            int n = Integer.parseInt(br.readLine());
            int[] arr = new int[n];
            StringTokenizer st = new StringTokenizer(br.readLine());

            for (int i = 0; i < n; i++) {
                arr[i] = Integer.parseInt(st.nextToken());
            }

            int goodPrefixCount = 0;
            long prefixSum = 0;

            for (int i = 0; i < n; i++) {
                prefixSum += arr[i];
                for (int j = 0; j <= i; j++) {
                    if (2 * arr[j] == prefixSum) {
                        goodPrefixCount++;
                        break;
                    }
                }
            }

            sb.append(goodPrefixCount).append("\n");
        }

        System.out.print(sb.toString());
    }
}

**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 edge cases and scenarios where the code might fail. The code iterates through the array and checks if any element in the current prefix is equal to half of the prefix sum. This approach can fail in several scenarios:

1. **Large Prefix Sum**: If the prefix sum becomes very large, the code might have issues with integer overflow or performance.
2. **Multiple Good Prefixes**: The code might not correctly handle cases where multiple elements in the prefix can be represented as the sum of the other elements.
3. **Edge Cases with Zeros**: The code might not handle cases with multiple zeros correctly.

Given these points, let's create a test input that includes these edge cases:

1. An array with multiple zeros.
2. An array where multiple elements can be represented as the sum of the other elements.
3. An array with a large prefix sum.

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

```plaintext
3
5
0 0 0 0 0
6
1 2 3 6 0 9
4
1000000000 1000000000 1000000000 3000000000
```

### Explanation:
1. **Test Case 1**: `[0, 0, 0, 0, 0]`
   - All prefixes are good because the sum of all elements in any prefix is zero, and each element is zero.
   - Expected output: `5`

2. **Test Case 2**: `[1, 2, 3, 6, 0, 9]`
   - The prefixes `[1, 2, 3, 6]` and `[1, 2, 3, 6, 0]` are good because `6 = 1 + 2 + 3` and `0 = 1 + 2 + 3 + 6`.
   - Expected output: `2`

3. **Test Case 3**: `[1000000000, 1000000000, 1000000000, 3000000000]`
   - The prefix `[1000000000, 1000000000, 1000000000, 3000000000]` is good because `3000000000 = 1000000000 + 1000000000 + 1000000000`.
   - Expected output: `1`

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
void solve(){
int n;
cin >> n ;
vector<int> a(n);
for(int i = 0; i < n ; i++) cin >> a[i];
long long sum = a[0];
int cnt = 0;
if(sum == 0)cnt++;
for(int i = 1 ; i < n ; i++){
    sum+=a[i];
    sort(a.begin(), a.begin() + i+1);
    sum -= a[i];
    if(sum == a[i]) cnt++;
    sum +=a[i];
}
cout << cnt << endl;

}

int main(){
int t;
cin >> t;
for(int i = 0; i < t; i++){
solve();
}
return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code might fail. The main issues in the buggy code are:

1. **Sorting the array repeatedly**: This is inefficient and unnecessary. It also changes the order of the elements, which can lead to incorrect results.
2. **Incorrect handling of the sum and prefix checks**: The code incorrectly updates the sum and checks the prefix without maintaining the correct state.

Given these issues, we can create a test case that:

1. **Has multiple elements that can form a good prefix**: This will test the correctness of the sum and prefix checks.
2. **Includes elements that are zero**: This will test the handling of zero elements.
3. **Includes elements that are not in a simple arithmetic sequence**: This will test the handling of more complex arrays.

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

```plaintext
3
5
1 0 1 2 4
4
0 0 0 0
6
1 2 3 6 0 12
```

Explanation:
1. **Test Case 1**: The array `[1, 0, 1, 2, 4]` has multiple good prefixes:
   - Prefix `[1]` is not good.
   - Prefix `[1, 0]` is good (sum of other elements is 1).
   - Prefix `[1, 0, 1]` is good (sum of other elements is 2).
   - Prefix `[1, 0, 1, 2]` is good (sum of other elements is 4).
   - Prefix `[1, 0, 1, 2, 4]` is good (sum of other elements is 4).
   - Expected output: `4`.

2. **Test Case 2**: The array `[0, 0, 0, 0]` has all prefixes good:
   - Prefix `[0]` is good (sum of other elements is 0).
   - Prefix `[0, 0]` is good (sum of other elements is 0).
   - Prefix `[0, 0, 0]` is good (sum of other elements is 0).
   - Prefix `[0, 0, 0, 0]` is good (sum of other elements is 0).
   - Expected output: `4`.

3. **Test Case 3**: The array `[1, 2, 3, 6, 0, 12]` has multiple good prefixes:
   - Prefix `[1]` is not good.
   - Prefix `[1, 2]` is not good.
   - Prefix `[1, 2, 3]` is good (sum of other elements is 6).
   - Prefix `[1, 2, 3, 6]` is good (sum of other elements is 6).
   - Prefix `[1, 2, 3, 6, 0]` is good (sum of other elements is 12).
   - Prefix `[1, 2, 3, 6, 0, 12]` is good (sum of other elements is 12).
   - Expected output: `4`.

This test input should effectively expose the issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Good Prefixes
time_limit_ms: 2000
memory_limit_mb: 256
Description: Alex thinks some array is good if there exists some element that can be represented as the sum of all other elements (the sum of all other elements is $0$ if there are no other elements). For example, the array $[1,6,3,2]$ is good since $1+3+2=6$. Furthermore, the array $[0]$ is also good. However, the arrays $[1,2,3,4]$ and $[1]$ are not good.

Alex has an array $a_1,a_2,\ldots,a_n$. Help him count the number of good non-empty prefixes of the array $a$. In other words, count the number of integers $i$ ($1 \le i \le n$) such that the length $i$ prefix (i.e. $a_1,a_2,\ldots,a_i$) is good.

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

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

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

Output: For each test case, output a single integer — the number of good non-empty prefixes of the array $a$.

Examples:
input:
7
1
0
1
1
4
1 1 2 0
5
0 1 2 1 4
7
1 1 0 3 5 2 12
7
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 294967296
10
0 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 589934592
output:
1
0
3
3
4
1
2

Note: In the fourth test case, the array has five prefixes:

  * prefix $[0]$ is a good array, as mentioned in the statement;    * prefix $[0, 1]$ is not a good array, since $0 \ne 1$;    * prefix $[0, 1, 2]$ is not a good array, since $0 \ne 1 + 2$, $1 \ne 0 + 2$ and $2 \ne 0 + 1$;    * prefix $[0, 1, 2, 1]$ is a good array, since $2 = 0 + 1 + 1$;    * prefix $[0, 1, 2, 1, 4]$ is a good array, since $4 = 0 + 1 + 2 + 1$.

As you can see, three of them are good, so the answer is $3$.

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

// Common macros
#define fast_io ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define ll long long
#define pqmin priority_queue<int, vector<int>, greater<int>>
#define pqmax priority_queue<int>
#define pb push_back
#define mp make_pair
#define F first
#define S second
#define all(v) v.begin(), v.end()
#define rall(v) v.rbegin(), v.rend()
#define rep(i, a, b) for (int i = a; i < b; ++i)
#define per(i, a, b) for (int i = a; i > b; --i)
#define repk(i, a, b, k) for (int i = a; i < b; i += k)
#define perk(i, a, b, k) for (int i = a; i > b; i -= k)
#define debug(x) cout << x << '\n';
#define debug_vec(v) do { cout << #v << " = "; for(auto x : v) cout << x << ' '; cout << '\n'; } while(0)
#define debug_pair(p) cout << #p << " = (" << p.F << ", " << p.S << ")\n";
#define lcm(a, b) ((a) / __gcd(a, b) * (b))
#define sq(x) ((x) * (x))
#define inf (int)1e9
#define linf (ll)1e18
#define mini(a, b) ((a) < (b) ? (a) : (b))
#define maxi(a, b) ((a) > (b) ? (a) : (b))
#define bitcount(x) __builtin_popcount(x)
#define bitcountll(x) __builtin_popcountll(x)
#define bitscan(x) __builtin_ctz(x)
#define bitscanll(x) __builtin_ctzll(x)
#define bitscanrev(x) __builtin_clz(x)
#define bitscanrevll(x) __builtin_clzll(x)
#define sum(v) accumulate(all(v), 0LL)
#define average(v) (sum(v) / v.size())
#define range(i, n) iota(i.begin(), i.end(), n)

int main() {
    fast_io;

    #ifndef ONLINE_JUDGE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    #endif

    int t;
    cin >> t;
    while (t--) {
        int n;
        cin>>n;
        vector<ll>arr(n);
        rep(i,0,n)cin>>arr[i];
        ll cnt=0;

        vector<ll>temp;
        rep(i,0,n){
            temp.push_back(arr[i]);
            ll max_ele=LONG_LONG_MIN;
            rep(i,0,temp.size()){
                if(temp[i]>max_ele)max_ele=temp[i];
            }
            ll sum=0;
            for(auto x : temp)sum+=x;
            if((sum-max_ele)==max_ele){
                ++cnt;
            }
        }
        cout<<cnt<<endl;
    }
    return 0;
}

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

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

Think step by step.

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

**Explanation:**
This test case is designed to challenge the buggy code by creating a scenario where the same value appears multiple times in the array. The array `[1, 1, 2, 2]` has the following prefixes:

1. `[1]` - Not good (1 is not the sum of an empty set).
2. `[1, 1]` - Not good (1 is not the sum of 1).
3. `[1, 1, 2]` - Good (2 is the sum of 1 + 1).
4. `[1, 1, 2, 2]` - Good (2 is the sum of 1 + 1 + 2).

The correct output for this test case should be `2`, as there are two good prefixes. The buggy code, however, might fail to correctly identify the good prefixes due to its inefficient and incorrect logic for checking the sum and maximum element.