DE-BUNDESWETTBEWERB 2003 Problem 6

The sequence $\{a_1,a_2,\ldots\}$ is recursively defined by $a_1 = 1$ , $a_2 = 1$ , $a_3 = 2$ , and $$a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0.$$ Prove that all elements of the sequence are integers.