
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: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$) passes through point A(1, $\frac{3}{2}$), with an eccentricity of $\frac{1}{2}$, and left and right focal points $F\_1$, $F\_2$ 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 $F\_2$AB 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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