
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
Given the ellipse $C\_1$: $\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1 (0 < b < 2)$ with an eccentricity equal to $\frac{\sqrt{3}}{2}$, and the parabola $C\_2$: $x^{2} = 2py (p > 0)$ with its focus on the vertex of ellipse $C\_1$.

1. Find the equation of the parabola $C\_2$.
2. Let $M(x\_1, y\_1)$ and $N(x\_2, y\_2)$ be two moving points on the parabola $C\_2$, where $y\_1 \neq y\_2$ and $y\_1 + y\_2 = 4$. The perpendicular bisector of line segment $MN$ intersects the $y$-axis at point $P$. Find the maximum area of $\triangle MNP$.

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