Example 7 Let the equation of the ellipse be $\frac{x^{2}}{100}+\frac{y^{2}}{36}=1, P\left(\frac{25}{4}, \frac{3 \sqrt{39}}{4}\right), F_{1}, F_{2}$ are the foci, find the equation of the incircle of $\triangle P F_{1} F_{2}$.