HMMT 2021 U23

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2 n) f(2 n+1)=$ $9 f(n)^{2}+3 f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$.