
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
Consider the right focus $F$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\,(a > 0,b > 0)$. A line perpendicular to the $x$-axis is drawn through $F$, intersecting the hyperbola and its asymptote at points $M$ and $N$ (both in the first quadrant), respectively. If $\overrightarrow{FM}=4\,\overrightarrow{MN}$, find the eccentricity of the hyperbola.

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