
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 $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with eccentricity $\sqrt{5}$, where $F_1$ and $F_2$ are the left and right foci of the hyperbola $C$. A line passing through point $F_{2}$ and perpendicular to the x-axis intersects the hyperbola $C$ above the x-axis at point $M$. Find the value of $\tan \angle MF_{1}F_{2}$.

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