
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
Point \( P \) is an arbitrary point on the right branch of the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\). Let \( F_{1} \) and \( F_{2} \) be the left and right foci, respectively. Define \(\angle P F_{1} F_{2} = \alpha\) and \(\angle P F_{2} F_{1} = \beta \). Express \(\frac{\tan \frac{\alpha}{2}}{\tan \frac{\beta}{2}}\) in terms of \(a\) and \(b\).

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