Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.) Proposed by Serbia