
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 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$, where $F$ is the right focus of the hyperbola. A perpendicular line is drawn from $F$ to one of its asymptotes, with the foot of the perpendicular being $Q$. The line $FQ$ intersects the left and right branches of the hyperbola at points $M$ and $N$, respectively. If $|MQ|=3|QN|$, then the eccentricity of the hyperbola is ____.

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