
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
Consider the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with left and right foci $F\_1$, $F\_2$, and an eccentricity of $\frac{1}{2}$. Let $P$ be a moving point on the ellipse such that the maximum area of $\triangle F\_1PF\_2$ is $\sqrt{3}$.
(1) Find the equation of the ellipse;
(2) Given that the line $l$ intersects the ellipse at points $A$ and $B$, and the equation of line $l$ is $y=kx+\sqrt{3} (k > 0)$, find the maximum area of $\triangle OAB$, where $O$ is the origin.

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