
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 $\frac{x^2}{2}+y^2=1$ with left and right foci at F<sub>1</sub> and F<sub>2</sub>, respectively, line l passes through the right focus F<sub>2</sub> and intersects the ellipse at points A and B.
(1) When the slope of line l is 1, and P is a moving point on the ellipse, how many points P satisfy the condition that the area of triangle ABP is $\frac{2\sqrt{5}-2}{3}$? Explain your reasoning.
(2) Does the area of the incircle of triangle ABF<sub>1</sub> have a maximum value? If it exists, find this maximum value and the equation of line l at this time; if it does not exist, explain the reason.

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