Problem: There is a string of length n+1 of characters 'A' and 'B'. The first character and last character of the string are equal to 'A'.

You are given m indices p_1, p_2, …, p_m (0-indexation) denoting the other indices of characters 'A' in the string.

Let's denote the minimum distance between two neighboring 'A' as l, and maximum distance between neighboring 'A' as r.

For example, (l,r) of string "ABBAABBBA" is (1,4).

And let's denote the balance degree of a string as the value of r-l.

Now Dreamoon wants to change exactly k characters from 'B' to 'A', and he wants to make the balance degree of the string as small as possible.

Please calculate the required minimum possible value of balance degree.

Input

The first line contains one integer t denoting the number of test cases (1 ≤ t ≤ 400 000).

For each test case, the first line contains three integers n, m and k (1 ≤ n ≤ 10^{15}, 0 ≤ m ≤ 400 000, 0 ≤ k < n - m).

The second line contains m integers p_1, p_2, …, p_m, (0 < p_1 < p_2 < … < p_m < n).

The total sum of m is at most 400 000.

Output

For each test case, print one integer: the smallest possible value of balance degree after k changes of 'B' to 'A'.

Example

Input


5
80 3 5
11 24 50
81 7 12
4 10 17 26 37 48 61
25 10 14
3 4 7 12 13 15 17 19 21 23
1 0 0

10 2 0
2 4


Output


5
2
0
0
4

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
using cat = long long;
int main() {
  cin.sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(10);
  int T;
  cin >> T;
  while (T--) {
    cat N, M, K;
    cin >> N >> M >> K;
    K += M + 1;
    vector<cat> A(M + 1);
    cat last = 0, cur;
    for (int i = 0; i <= M; i++) {
      if (i < M)
        cin >> cur;
      else
        cur = N;
      A[i] = cur - last;
      last = cur;
    }
    if (N % K == 0) {
      cat v = N / K;
      bool ok = true;
      for (int i = 0; i <= M; i++)
        if (A[i] % v) ok = false;
      if (ok) {
        cout << "0\n";
        continue;
      }
    }
    cat l_mi = 0, l_mx = N + 1;
    while (l_mx - l_mi > 1) {
      cat l = (l_mi + l_mx) / 2, val = 0;
      for (int i = 0; i <= M; i++) val += A[i] / l;
      if (val >= K)
        l_mi = l;
      else
        l_mx = l;
    }
    for (int i = 0; i <= M; i++) l_mi = min(l_mi, A[i]);
    cat r_mi = 0, r_mx = N + 1;
    while (r_mx - r_mi > 1) {
      cat r = (r_mi + r_mx) / 2, val = 0;
      for (int i = 0; i <= M; i++) val += (A[i] + r - 1) / r;
      if (val <= K)
        r_mx = r;
      else
        r_mi = r;
    }
    cat l0 = l_mi, r0 = r_mx;
    vector<pair<cat, cat> > LR;
    for (int i = 0; i <= M; i++)
      if ((A[i] + r0 - 1) / r0 > A[i] / l0) {
        cat v = (A[i] + r0 - 1) / r0;
        l_mi = 0, l_mx = l0;
        while (l_mx - l_mi > 1) {
          cat l = (l_mi + l_mx) / 2;
          if (A[i] / l >= v)
            l_mi = l;
          else
            l_mx = l;
        }
        v = A[i] / l0;
        r_mi = r0, r_mx = max(r0, A[i]) + 1;
        while (r_mx - r_mi > 1) {
          cat r = (r_mi + r_mx) / 2;
          if ((A[i] + r - 1) / r <= v)
            r_mx = r;
          else
            r_mi = r;
        }
        LR.push_back({l_mi, r_mx});
      }
    if (LR.empty()) {
      cout << r0 - l0 << "\n";
      continue;
    }
    sort(begin(LR), end(LR));
    cat D = r0 - begin(LR)->first, r = r0;
    for (int i = 0; i < (int)LR.size(); i++) {
      r = max(r, LR[i].second);
      cat l = l0;
      if (i + 1 < (int)LR.size()) l = LR[i + 1].first;
      D = min(D, r - l);
    }
    cout << D << "\n";
  }
  return 0;
}