Collinear Points on Twin Conics

Let $A_1,A_2$ be the left and right vertices of the ellipse $C_1:\dfrac{x^2}3+\dfrac{y^2}{b^2}=1$ ($0<b<\sqrt 3$). Let $M$ be a point on $C_1$ and $N$ a point on the hyperbola $C_2:\dfrac{x^2}3-\dfrac{y^2}{b^2}=1$ (with $M,N$ distinct from $A_1,A_2$). Let $k_1,k_2,k_3,k_4$ be the slopes of $A_1M$, $A_2M$, $A_1N$, $A_2N$, with $k_1+k_2+k_3+k_4=0$.

1. Let $O$ be the origin. Prove that $O,M,N$ are collinear. 2. Let $F_1$, $F_2$ be the right foci of $C_1$, $C_2$ respectively, with $M$, $N$ both in the first quadrant; the lines $NF_1$ and $MF_2$ meet at a point $P$, and $k_1^2+k_2^2+k_3^2+k_4^2=8$. (a) Prove that $MF_1\parallel NF_2$. (b) Prove that $\angle A_1PF_1=\angle A_2PF_2$.