
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$, with an eccentricity of $\frac{1}{2}$. The left and right foci of the ellipse are $F_1$ and $F_2$ respectively. If point $M$ is a movable point on the ellipse, find the equation of the ellipse $C$ and the maximum value of the area of triangle $\triangle MF_1F_2$ which is given as $\sqrt{3}$.

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