A Triangle Inscribed in One Parabola, Tangent to Another

As shown in the figure, the triangle $ABC$ is inscribed in the parabola $E: x^2 = y$, and the lines containing the sides $AB$ and $AC$ are both tangent to the parabola $M: y^2 = 4x$. Let $F$ be the focus of $M$.

1. Prove that the line containing the side $BC$ is tangent to the parabola $M$. 2. Prove that the points $A$, $C$, $B$, $F$ lie on a common circle.