
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 two foci $F_{1}$ and $F_{2}$, eccentricity $\frac{1}{2}$, a line $l$ passing through $F_{1}$ intersects ellipse $C$ at points $M$ and $N$, and the perimeter of $\triangle MNF_{2}$ is $8$.<br/>$(1)$ Find the equation of ellipse $C$;<br/>$(2)$ Find the maximum area of $\triangle MNF_{2}$.

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