
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\ (a > b > 0)$ with an eccentricity of $\frac{1}{2}$, a tangent line $l$ passing through point $P(0, \frac{\sqrt{3}}{2})$ intersects the ellipse at points $A$ and $B$. When line $l$ is parallel to the $x$-axis, the length of the line segment cut off by the ellipse is $2\sqrt{3}$. $(F_{1},F_{2})$ are the left and right foci, respectively.
(1) Find the standard equation of the ellipse;
(2) A line $l'$ passing through $F_{2}$ intersects the ellipse at two distinct points $M$ and $N$. Does the inscribed circle of triangle $F_{1}MN$ have a maximum area? If so, find the maximum area and the equation of line $l'$ at this time. If not, explain the reason.

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