
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
Let \( P \) be the intersection point of the directrix \( l \) of an ellipse and its axis of symmetry, and \( F \) be the corresponding focus. \( AB \) is a chord passing through \( F \). Find the maximum value of \( \angle APB \) which equals \( 2 \arctan e \), where \( e \) is the eccentricity.

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