
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through point $A(2,3)$, and its eccentricity $e = \frac{1}{2}$.

$(1)$ Find the equation of the ellipse $E$;  
$(2)$ If the angle bisector of $\angle F_1AF_2$ lies on line $l$ and intersects the ellipse $E$ at another point $B$, and $C$ is a point on the ellipse $E$, when the area of $\triangle ABC$ is maximized, find the coordinates of point $C$.

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