2016 AIME II Problem 5

Triangle ABC_0 has a right angle at C_0. Its side lengths are pairwise relatively prime positive integers, and its perimeter is p. Let C_1 be the foot of the altitude to \overline{AB}, and for n \geq 2, let C_n be the foot of the altitude to \overline{C_{n-2}B} in \triangle C_{n-2}C_{n-1}B. The sum \sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p. Find p.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.