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

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

Let $$S$$ be the set of all functions $$f:A\to B$$ such that $$f(2)\neq 2$$ and $$f(4)\neq 4$$. Consider the set

$$T=\big\{f\in S:\text{there exists a function }g:B\to\mathbb{N}\text{ such that }g\big(f(x)\big)=2^x\text{ for all }x\in A\big\}.$$

Then the number of elements in the set $$T$$ is ___.