
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 left and right foci $F_{1}$ and $F_{2}$, an eccentricity $e=\frac{1}{2}$, and a minor axis length of $2\sqrt{3}$.

(1) Find the standard equation of the ellipse $C$.
(2) If a line $l$ passes through $F_{2}$ and intersects the ellipse $C$ at two distinct points $A$ and $B$, determine whether the area of $\triangle F_{1}AB$ has a maximum value. If it does, find this maximum value and the equation of the line $l$; if not, explain why.

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