Laura is putting together the following list: $a_0, a_1, a_2, a_3, a_4, ..., a_n$, where $a_0 = 3$ and $a_1 = 4$. She knows that the following equality holds for any value of $n$ integer greater than or equal to $1$: $$a_n^2-2a_{n-1}a_{n+1} =(-2)^n.$$Laura calculates the value of $a_4$. What value does it get?