Functions

Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets

\(A=\{1,2,3,4,5\} \text { and } B=\{1,2,3,4,5,6,7\} .\)

Let $S$ be the set of all functions $f: A \rightarrow B$ such that $f(2) \neq 2$ and $f(4) \neq 4$. Consider the set $T=\left\{f \in S:\right.$ there exists a function $g: B \rightarrow \mathbb{N}$ such that $g(f(x))=2^x$ for all $\left.x \in A\right\}$.

Then the number of elements in the set $T$ is $\_\_\_\_$ .