
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 a hyperbola $C$ with eccentricity $\sqrt{5}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$. The line $l: y=4x-3a-1$ intersects the right branch of $C$ at points $A$ and $B$, and intersects the two asymptotes of $C$ at points $M$ and $N$, with $M$, $A$, $B$, $N$ arranged from top to bottom such that $|MA|=\frac{1}{2}|AB|$.
1. Find the equation of the hyperbola $C$.
2. Find the area of triangle $AOM$.

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