Problem: It is the hard version of the problem. The only difference is that in this version 3 ≤ k ≤ n.

You are given a positive integer n. Find k positive integers a_1, a_2, …, a_k, such that:

  * a_1 + a_2 + … + a_k = n 
  * LCM(a_1, a_2, …, a_k) ≤ n/2 



Here LCM is the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of numbers a_1, a_2, …, a_k.

We can show that for given constraints the answer always exists.

Input

The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of test cases.

The only line of each test case contains two integers n, k (3 ≤ n ≤ 10^9, 3 ≤ k ≤ n).

It is guaranteed that the sum of k over all test cases does not exceed 10^5.

Output

For each test case print k positive integers a_1, a_2, …, a_k, for which all conditions are satisfied.

Example

Input


2
6 4
9 5


Output


1 2 2 1 
1 3 3 1 1 

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
#include <string>
#include <vector>
#include <queue>
#include <map>
#include <set>
#include <deque>
#include <math.h>
using namespace std;

int main()
{
  ios_base::sync_with_stdio(false);
  cin.tie(NULL);
  long long z;
  cin >> z;
  //z = 1;
  while (z--)
  {
    long long int k = 0,
                  x = 0, y, z, s0 = 0, s1 = 0, s2 = 0, l, r, u,
                  s3 = 0, s5 = 0, s6 = 0, s4 = 0, p1 = 0, p2 = 0,
                  p3 = 0, ans = 0, mid = 0, n, m, i = 0, j = 0;
    cin >> n >> m;
    for (i = 1; i <= m - 3; i++)
      cout << "1 ";
    s1 = (n - (m - 3));
    n = s1;
    if (n % 2 != 0)
      cout << "1 " << (n - 1) / 2 << ' ' << (n - 1) / 2 << "\n";
    else
    {
      if ((n / 2) % 2 != 0)
        cout << "2 " << (n - 2) / 2 << ' ' << (n - 2) / 2 << "\n";
      else
        cout << n / 2 << ' ' << (n / 2) / 2 << ' ' << (n / 2) / 2 << "\n";
    }
  }
}