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.