
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
An ellipse $C$ with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through point $A(1, \frac{3}{2})$, with an eccentricity of $\frac{1}{2}$. The left and right foci are $F_1$ and $F_2$, respectively. A line passing through $F_1$ intersects the ellipse at points $A$ and $B$.

(I) Find the equation of ellipse $C$;

(II) When the area of triangle $F_2AB$ is $\frac{12\sqrt{2}}{7}$, find the equation of the line.

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