
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, eccentricity $e = \frac{1}{2}$, and it passes through the point $(-1, \frac{3}{2})$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ A line $l: y = x + m$ is tangent to the ellipse $C$. Points $M$ and $N$ are on line $l$, and $F_1M \perpendicular l$, $F_2N \perpendicular l$. Find the area of the quadrilateral $F_1MNF_2$.

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