
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 an ellipse centered at the origin $O$ with its foci on the $x$-axis, an eccentricity of $e=\frac{1}{2}$, and passing through the point $\left(1,\frac{3}{2}\right)$.

(1) Find the equation of the ellipse;

(2) Denote the left and right foci of the ellipse as $F\_1$ and $F\_2$, respectively. A line $l$ passes through $F\_2$ and intersects the ellipse at two distinct points $A$ and $B$. Does the inscribed circle of $\triangle F\_1AB$ have a maximum area? If so, find the maximum value and the equation of the line when it occurs. If not, explain the reason.

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