CN-LEAGUE 2020 Problem 3

Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$