GAIC Math 2024 Problem 64

Given hyperbola $\Gamma: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes the point $M(3,\sqrt{2})$, line $l$ passes its right focus $F(2,0)$ and cross the right branch of $\Gamma$ at points $A$ and $B$, and cross the y-axis at point P. If $\overrightarrow{PA}=m\overrightarrow{AF}, \overrightarrow{PB}=n\overrightarrow{BF}$, then $m+n=$ _____.