
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, there is an ellipse C given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$). The left and right foci of the ellipse are $F_1$ and $F_2$, respectively, and the eccentricity is $\frac{\sqrt{3}}{2}$. A circle centered at the origin with radius equal to the short semi-axis of ellipse C 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 ellipse C;  
(2) If $\overrightarrow{MF_{1}} = 3\overrightarrow{F_{1}N}$, find the equation of line $l$;  
(3) Find the maximum area of triangle $F_1MN$.

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