
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with eccentricity $e$.  
$(1)$ If $e = \sqrt{2}$ and the hyperbola $E$ passes through the point $(\sqrt{2}, 1)$, find the equation of the hyperbola $E$.  
$(2)$ If $a = 2$, the left and right foci of the hyperbola $E$ are $F_{1}$ and $F_{2}$ respectively. The distance from the foci to the asymptotes of the hyperbola is $\sqrt{3}$. Point $M$ is in the first quadrant and on the hyperbola $E$. If $|MF_{1}| = 8$, find the value of $\cos \angle F_{1}MF_{2}$.  
$(3)$ Let $O: x^{2} + y^{2} = 4$, $k$, $m \in \mathbb{R}$. If the moving line $l: y = kx + m$ is tangent to the circle $O$ and intersects the hyperbola $E$ at points $A$ and $B$ such that $\angle AOB = \frac{\pi}{2}$, find the range of eccentricity $e$ of the hyperbola $E$.

Present the answer in LaTex format: \boxed{Your answer}
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