
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
Point \( P \) is an arbitrary point on the ellipse \(\frac{(x+4)^{2}}{9}+\frac{y^{2}}{16}=1\). Let \( P A \) and \( P B \) be the two tangents to the parabola \( y^{2}=4x \), with \( A \) and \( B \) being the points of tangency. Find the maximum area of the triangle \( \triangle P A B \) and specify the coordinates of point \( P \) when the area of \( \triangle P A B \) is maximized.

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