
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
Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. Given $\angle P F_{1} F_{2}=\alpha$ and $\angle P F_{2} F_{1}=\beta$, express $\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}$ in terms of $e$.

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