Find a non-constant real-analytic function $f(x)$ such that for $x\in\Bbb R,\;f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Question

Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies :

$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Before you ask if this simplifies by writing $2^x = y$ note that $2^x$ is never equal to $0$.

I think $f(x)$ is unique upto a multiplication constant $C$.

How to find $f(x)$ ?

What would make an excellent asymptotic to $f(x)$ ?