The equation of an ellipse is \[\frac{x^2}{2} + y^2 = 1.\] Let $F = (1,0).$ There exists a point $P = (p,0),$ where $p > 0,$ such that for any chord $\overline{AB}$ which contains $F,$ $\overline{FP}$ passes through the point $(-1,0)$. Find $p.$
