What Are Recursive Sequences?

Let $a_1,a_2,...$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integer $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.

(Source: USAMTS, Year 25, Round 3, Problem 2)