It is the hard version of the problem. The only difference is that in this version $3 \le k \le n$.

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

• $a_1 + a_2 + \ldots + a_k = n$
• $LCM(a_1, a_2, \ldots, a_k) \le \frac{n}{2}$

Here $LCM$ is the least common multiple of numbers $a_1, a_2, \ldots, a_k$.

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

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

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

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

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