
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 $a > b > 0$, where $F_{1}$ and $F_{2}$ are the left and right foci of $\Gamma$ respectively, and $|F_{1} F_{2}|=2c$. Let $A$ be the right endpoint of $\Gamma$, and let the line $l$ pass through point $A$ and be perpendicular to the $x$-axis. If $P$ is a moving point on line $l$ such that the maximum value of $\angle F_{1} P F_{2}$ is $\frac{\pi}{4}$, what are the coordinates of point $P$?

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