Product Sum Arrays (Hard)

**This is the hard version of the problem. Here, you need to find the exact value of $R$.** You are given an integer $N$. An array $A$ is called *good* if it satisfies all of the following conditions: - $1 \le A_i \le N$ for all $1 \le i \le |A|$ - $A_i \le A_{i + 1}$ for all $1 \le i < |A|$ - $A_1 + A_2 + \ldots + A_{|A|} = N$ - $A_1 \times A_2 \times \ldots \times A_{|A|} = N$ In other word