
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 a minor axis length of $2\sqrt{3}$ and eccentricity $e=\frac{1}{2}$, <br/>$(1)$ Find the standard equation of ellipse $C$. <br/>$(2)$ If $F_{1}$ and $F_{2}$ are the left and right foci of ellipse $C$ respectively, and the line $l$ passing through $F_{2}$ intersects ellipse $C$ at two distinct points $A$ and $B$, find the maximum area of $\triangle F_{1}AB$.

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