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

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