KöMaL 2013 A598

Denote by $u_n$ the $n$th Fibonacci number ($u_1$=$u_2$=1, $u_{n+1}$=$u_n$+$u_{n-1}$). Prove that if $a$,$b$,$c$>1 are integers such that $a$ divides $u_b$, $b$ divides $u_c$ and $c$ divides $u_a$, then 5 divides $a$, $b$ and $c$, or 12 divides $a$, $b$and $c$.