
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a \gt b \gt 0)$ with left and right foci $F_{1}$ and $F_{2}$, and $M$ as a moving point on the ellipse $C$, where the maximum value of $\angle F_{1}MF_{2}$ is $120^{\circ}$, and the maximum value of the distance from point $M$ to the right focus $F_{2}$ is $2+\sqrt{3}$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ Given a line $l$ passing through point $F_{2}$ intersects the ellipse $C$ at points $A$ and $B$, find the equation of line $l$ when the area of $\triangle F_{1}AB$ is maximized.

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