
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 an ellipse \(\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \(a > b > 0\). The left and right endpoints of the major axis are \(A\) and \(B\), respectively, and the left and right foci are \(F_{1}\) and \(F_{2}\), respectively. If there exists a point \(P\) on the ellipse \(\Gamma\) such that \(P F_{1}\) and \(P F_{2}\) trisect \(\angle APB\), find the number of distinct eccentricities \(e\) of the ellipse \(\Gamma\) that satisfy this condition.

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