
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 that \( P \) is a point on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), \( F_{1} \) and \( F_{2} \) are the two foci of the ellipse, and \( e \) is the eccentricity of the ellipse. If \(\angle P F_{1} F_{2}=\alpha \) 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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