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 JAVA):

Solution:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.StringTokenizer;

public class problemC2 {
    static class Solution {
        void solve() {
            int n = fs.nextInt();
            int k = fs.nextInt();

            int[] ans = new int[k];
            for (int i = 3 ; i < k ; i ++ ) { ans[i] = 1; n -- ; }
            int[] a = solve3(n);
            for (int i = 0 ; i < 3 ; i ++ ) ans[i] = a[i];
            print(ans);
        }

        int[] solve3(int n) {
            int[] a = try0(n);
            if (a != null) return a;
            a = try1(n);
            if (a != null) return a;
            a = try2(n);
            return a;
        }

        int[] try0(int n) {
            if (n % 3 == 0) {
                int x = n/3;
                return new int[]{x, x, x};
            }
            return null;
        }

        int[] try2(int n) {
            int rem = n-2;

            if (rem % 2 == 1) return null;
            int x = rem / 2;
            if (x % 2 == 0) return new int[]{2, x, x};
            if (x == 1) return new int[]{2, 1, 1};
            int[] a = try2(n/2);
            for (int i = 0 ; i < 3 ; i ++ ) a[i] *= 2;
            return a;
        }

        void print(int[] a) {
            for (int x : a) out.print(x + " ");
            out.println();
        }

        int[] try1(int n) {
            int rem = n-1;
            if (rem % 2 == 1) return null;
            rem /= 2;
            return new int[]{1, rem, rem };
        }
    }

    public static void main(String[] args) throws Exception {
        int T = 1;
        T = fs.nextInt();
        Solution solution = new Solution();
        for (int t = 0; t < T; t++) solution.solve();
        out.close();
    }

    static void debug(Object... O) {
        System.err.println("DEBUG: " + Arrays.deepToString(O));
    }

    private static FastScanner fs = new FastScanner();
    private static PrintWriter out = new PrintWriter(System.out);

    static class FastScanner { // Thanks SecondThread.
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        StringTokenizer st = new StringTokenizer("");

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

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

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

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

        ArrayList<Integer> readArrayList(int n) {
            ArrayList<Integer> a = new ArrayList<>(n);
            for (int i = 0 ; i < n; i ++ ) a.add(fs.nextInt());
            return a;
        }

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

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

        String nextString() {
            return next();
        }
    }
}