For non-negative integers $s$ and $r$, let $\left(\begin{array}{l}s\\ r\end{array}\right)=\left\{\begin{array}{cc}\frac{s!}{r!\left(s-r\right)!}& \mathrm{if} r\le s,\\ 0& \mathrm{if} r>s.\end{array}\right.$

For positive integers $m$ and $n$, let $g\left(m,n\right)=\sum _{p=0}^{m+n}\frac{f\left(m,n,p\right)}{\left(\begin{array}{c}n+p\\ p\end{array}\right)}$

Where for any non-negative integer $p$, $f\left(m,n,p\right)=\sum _{i=0}^p\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n+i\\ p\end{array}\right)\left(\begin{array}{c}p+n\\ p-i\end{array}\right)$

Then which of the following statement is/are TRUE?