
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
In the rectangular coordinate system $xOy$, it is known that the eccentricity of the ellipse $(C\_1)$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b \geqslant 1)$ is $e = \frac{\sqrt{3}}{2}$, and the maximum distance from a point $M$ on the ellipse $(C\_1)$ to the point $Q(0, 3)$ is $4$.
(I) Find the equation of the ellipse $(C\_1)$;
(II) Let $A(0, \frac{1}{16})$, and $N$ be a moving point on the parabola $(C\_2)$: $y = x^{2}$. The tangent line to the parabola $(C\_2)$ at point $N$ intersects the ellipse $(C\_1)$ at points $B$ and $C$. Find the maximum area of $\triangle ABC$.

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