AMC 12 2024b Problem 21

The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?

$\textbf{(A) }40 \textbf{(B) }126 \textbf{(C) }154 \textbf{(D) }176 \textbf{(E) }208$