Inspired by Nihar Mahajan's featured problem

In each of the cases below, there exists only one solution whereby the sum of $n$ positive integers equals the product.

• For $n=2$, we have $2+2=2\times 2$.
• For $n=3$, we have $1+2+3=1\times2\times3$.
• For $n=4$, we have $1+1+2+4=1\times 1\times2\times4$.

If solution(s) exist for $n=5,$ find the sum of all possible sums (or equivalently, products).

If such a solution does not exist, submit your answer as 0.