
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 that the center of the hyperbola $C$ is at the origin, point $F(2,0)$ is one of the foci of the hyperbola $C$, and the perpendicular line $l$ to the asymptote passing through point $F$ has a foot $M$. The line $l$ intersects the $y$-axis at point $E$. If $|FM|=3|ME|$, find the equation of the hyperbola $C$.

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