
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 that the center of the ellipse C is at the origin, its eccentricity is equal to $\frac{1}{2}$, and one of its minor axis endpoints is the focus of the parabola x²=$4\sqrt{3}y$.
1. Find the standard equation of the ellipse C.
2. The left and right foci of the ellipse are $F_1$ and $F_2$, respectively. A line $l$ passes through $F_2$ and intersects the ellipse at two distinct points A and B. Determine whether the area of the inscribed circle of $\triangle{F_1AB}$ has a maximum value. If it exists, find this maximum value and the equation of the line at this time; if not, explain the reason.

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