2016 AIME I Problem 10

A strictly increasing sequence of positive integers a_1, a_2, a_3, \cdots has the property that for every positive integer k, the subsequence a_{2k-1}, a_{2k}, a_{2k+1} is geometric and the subsequence a_{2k}, a_{2k+1}, a_{2k+2} is arithmetic. Suppose that a_{13} = 2016. Find a_1.

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.