Counting Inscribed Incircle Points

Let $F_1,F_2$ be the left and right foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$. If $M$ is a point on the ellipse such that the incircle of $\triangle MF_1F_2$ has circumference $3\pi$, find the number of such points $M$.