
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
Ellipse $\(C\)$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has its minor axis endpoints as $B\_1(0, -1)$, $B\_2(0, 1)$, with an eccentricity of $e = \frac{\sqrt{3}}{2}$. Point $P$ is any point on the ellipse $C$ not on the coordinate axes. Lines $B\_1P$ and $B\_2P$ intersect the $x$-axis at points $M$ and $N$, respectively.

(I) Find the equation of ellipse $C$ and the value of $|OM| \cdot |ON|$;

(II) If the coordinate of point $M$ is $(1, 0)$, find the maximum area of triangle $ABN$, where line $I$ passing through point $M$ intersects ellipse $C$ at points $A$ and $B$.

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