Claims About a Diameter Circle and a Hyperbola

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$ and left and right vertices $A_1$, $A_2$. The circle with diameter $F_1F_2$ meets one asymptote of $C$ at points $M$ and $N$, with $\angle NA_1M=\dfrac{5\pi}6$. Determine, with proof, which of the following claims are true.

1. $\angle A_1MA_2=\dfrac{\pi}6$. 2. $|MA_1|=2|MA_2|$. 3. The eccentricity of $C$ is $\sqrt{13}$. 4. When $a=\sqrt 2$, the area of the quadrilateral $NA_1MA_2$ is $8\sqrt 3$.