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.


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