
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 coordinates of the foci of an ellipse are $F_1(-1,0)$ and $F_2(1,0)$, and one of the vertices on the minor axis, $B$, satisfies $\overrightarrow{BF_{1}} \cdot \overrightarrow{BF_{2}} = 2$.
(1) Find the equation of the ellipse;
(2) If a line $l$ passes through $F_2$ and intersects the ellipse at two distinct points $M$ and $N$, does the area of the incircle of triangle $F_1MN$ have a maximum value? If it exists, find this maximum value and the equation of the line at this time; if it does not exist, explain why.

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