
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 $\frac{1}{2}$. $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse C, respectively, and $|F_{1}F_{2}|=2$.
(1) Find the equation of the ellipse C;
(2) Let M be any point on the ellipse, and a circle M with center M and radius $MF_{1}$ is drawn. When the circle M intersects with the line $l: x= \frac{a^{2}}{c}$, find the maximum area of the triangle $MF_{1}F_{2}$.

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