
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
The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ has a right focus at $F$ and an eccentricity of $e$. A line passing through point $F$ with a slope angle of $\frac{\pi}{3}$ intersects this hyperbola at points $A$ and $B$. If the midpoint of $A B$ is $M$ and the distance $|FM|$ equals the semi-focal distance, find $e$.

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