JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 21

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $f:S\to S$ as
$$f(n)=\begin{cases}2n, & \text{if } n=1,2,3,4,5\\2n-11, & \text{if } n=6,7,8,9,10\end{cases}$$
Let $g:S\to S$ be a function such that $f\circ g(n)=\begin{cases}n+1, & \text{if } n \text{ is odd}\\n-1, & \text{if } n \text{ is even}\end{cases}$.

Then $g(10)\left(g(1)+g(2)+g(3)+g(4)+g(5)\right)$ is equal to ______.