
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
Consider an ellipse (C) with the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through the point $(1, \frac{3}{2})$. The eccentricity of the ellipse is $\frac{1}{2}$, and its left and right foci are $F\_1$ and $F\_2$, respectively. Let $A$ and $B$ be the intersection points of the ellipse with a line passing through $F\_1$.

(1) Find the equation of the ellipse (C);
(2) 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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