An Angle Bisector Intercept

Let $P$ be a point of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) in the first quadrant, and $F_1, F_2$ the left and right foci. The bisector of $\angle F_1PF_2$ meets the $y$-axis at a point with $y$-coordinate $t$. Find the range of possible values of $t$, in terms of $a$ and $b$.