
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 \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left focus \( F \), and a line \( l \) passing through \((1,1)\) that intersects the ellipse at points \( A \) and \( B \). When the perimeter of triangle \( \triangle FAB \) is at its maximum, what is the area of triangle \( \triangle FAB \)?

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