
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 ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), let \(F_{1}\) and \(F_{2}\) be the left and right foci of the ellipse, respectively. Let \(P\) be a point on the ellipse. Then, the excircle of \(\triangle P F_{1} F_{2}\) opposite to \(\angle P F_{1} F_{2}\) is tangent to the right vertex \(A_{2}\) of the ellipse, and the excircle of \(\triangle P F_{1} F_{2}\) opposite to \(\angle P F_{2} F_{1}\) is tangent to the left vertex \(A_{1}\) of the ellipse.


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