
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 \(\Gamma\):

\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a>b>0)
\]

with foci \(F_{1}\) and \(F_{2}\) such that \(\left|F_{1} F_{2}\right|=2\). Let \(A\) be the rightmost vertex of \(\Gamma\), and let the line \(l\) pass through point \(A\) and be perpendicular to the \(x\)-axis. Point \(P\) is a moving point on the line \(l\). If the maximum value of \(\angle F_{1} P F_{2}\) is \(\frac{\pi}{4}\), then find the coordinates of point \(P\).

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