
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 $F_{1}(-1,0)$, $F_{2}(1,0)$, and a line perpendicular to the major axis through $F_{2}$ intersects the ellipse at points $P$ and $Q$, with $|PQ|=3$.
$(1)$ Find the equation of the ellipse;
$(2)$ A line $l$ through $F_{2}$ intersects the ellipse at two distinct points $M$ and $N$. Does the area of the incircle of $\triangle F_{1}MN$ have a maximum value? If it exists, find this maximum value and the equation of the line at this time; if not, explain why.

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