Two Focal Chords of a Parabola and a Parallelogram

As shown in the figure, let $F$ be the focus of the parabola $y^2 = 4x$, and let $AB$ and $CD$ be two chords of the parabola passing through $F$. Let $P$ be the intersection point of lines $AC$ and $BD$, and let lines $AC$ and $BD$ meet the $y$-axis at $M$ and $N$ respectively.

1. Prove that $P$ always lies on the directrix of the parabola. 2. Prove that quadrilateral $PMFN$ is a parallelogram.