
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 $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$, and a point $(\sqrt{3},\frac{1}{2})$ on the ellipse $C$.

(1) Find the equation of the ellipse $C$;

(2) If a line $l$ intersects the ellipse $C$ at points $P$ and $Q$, and the midpoint of the segment $PQ$ is $H$, $O$ is the origin of the coordinate and $OH=1$, find the maximum value of the area of $\triangle POQ$.

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