CONO-SUR 2003 Problem 2

Define the sequence $\{a_n\}$ in the following manner: $a_1=1$ $a_2=3$ $a_{n+2}=2a_{n+1}a_{n}+1$ ; for all $n\geq1$ Prove that the largest power of $2$ that divides $a_{4006}-a_{4005}$ is $2^{2003}.$