
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the length of the minor axis being $2\sqrt{3}$ and the eccentricity $e = \frac{1}{2}$,  
(1) Find the standard equation of the ellipse $C$.  
(2) If $F_1$ and $F_2$ are the left and right foci of the ellipse $C$, respectively, and a line $l$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $A$ and $B$, find the maximum value of the area of $\triangle F_1AB$.

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