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Arithmetico-geometric... sequence?

Source: 2016 AIME I #10

March 4, 2016
AIME2016 AIME ISequence

Problem Statement

A strictly increasing sequence of positive integers a1,a2,a3,…a_1, a_2, a_3, \ldots has the property that for every positive integer kk, the subsequence a2k−1,a2k,a2k+1a_{2k-1}, a_{2k}, a_{2k+1} is geometric and the subsequence a2k,a2k+1,a2k+2a_{2k}, a_{2k+1}, a_{2k+2} is arithmetic. Suppose that a13=2016a_{13} = 2016. Find a1a_1.
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