If $(-1)^{\gcd(n,F_n)}=1$ where $F_n$ is the $n$-th Fibonacci number then $n$ is divisible by $6$ ?

Question

Question: If $n$ is any positive integer I write $F_0=0, F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. The numbers $F_n$ are called the Fibonacci numbers. Let $a_n=(-1)^{\gcd(n,F_n)}$. Is it true that if $a_n=1$ then $n$ is a multiple of $6$ ?

Not much motivation here just an experimental observation I noticed. I am not sure where exactly to start since I do not know a formula for $\gcd(n,F_n)$.