
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 $\frac{1}{2}$. The left and right foci of the ellipse are $F_{1}$ and $F_{2}$, respectively, and $M$ is a moving point on the ellipse. The maximum area of $\triangle MF_{1}F_{2}$ is $\sqrt{3}$.
(I) Find the equation of the ellipse $(C)$;
(II) Let $P$ be a point on the ellipse, with $PF_{1}$ intersecting the $y$-axis at $Q$, and $\overrightarrow{F_{1}P}=2\overrightarrow{F_{1}Q}$. If $PF_{1}$ intersects the ellipse at another point $R$, find the area of $\triangle PRF_{2}$.

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