
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
In the Cartesian coordinate system $xOy$, the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ are $F_1$ and $F_2$ respectively, with an eccentricity of $\frac{\sqrt{3}}{2}$. A circle centered at the origin with the minor axis length of the ellipse $C$ as its radius is tangent to the line $x - y + \sqrt{2} = 0$. A line $l$ passing through point $F_2$ intersects the ellipse at points $M$ and $N$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ If $\overrightarrow{MF_1} = 3\overrightarrow{F_1N}$, find the equation of the line $l$;  
$(3)$ Find the maximum area of $\triangle F_1MN$.

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