2023 Gauss 8 Problem 25

Given the list 2, 3, 4, 5, there are exactly three different ways to choose three integers from the list and form a triangle whose side lengths are equal to those integers. The integers chosen could be 2, 3, 4 or 2, 4, 5 or 3, 4, 5. The integers 2, 3, 5 cannot be used as side lengths of a triangle. Given the list 4, 10, 3, n, 13, there are exactly four different ways to choose three integers from the list and form a triangle whose side lengths are equal to those integers. If n is different from all other numbers in the list, then the sum of all possible values of n is

\textbf{(A)}\ 46\quad \textbf{(B)}\ 29\quad \textbf{(C)}\ 69\quad \textbf{(D)}\ 23\quad \textbf{(E)}\ 17

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.