13:I[87,["736","static/chunks/89fc4f0e-9e200ae6a0166583.js","401","static/chunks/363642f4-32903b9154b7835a.js","153","static/chunks/153-58eddeb032cc4608.js","131","static/chunks/131-05ce00cfd760e12b.js","87","static/chunks/87-b76b60eef573119a.js","286","static/chunks/app/problem/%5Bid%5D/page-45cf3875257fc32e.js"],"Header"] 14:I[1742,["736","static/chunks/89fc4f0e-9e200ae6a0166583.js","401","static/chunks/363642f4-32903b9154b7835a.js","153","static/chunks/153-58eddeb032cc4608.js","131","static/chunks/131-05ce00cfd760e12b.js","87","static/chunks/87-b76b60eef573119a.js","286","static/chunks/app/problem/%5Bid%5D/page-45cf3875257fc32e.js"],"ProblemView"] 15:T604,

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.

Input

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 $$$10^5$$$.

k-LCM (hard version)

Examples