
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 an ellipse $C_1: \frac{x^2}{4} + \frac{y^2}{b^2} = 1$ ($0 < b < 2$) with eccentricity equal to $\frac{\sqrt{3}}{2}$, and a parabola $C_2: x^2 = 2py$ ($p > 0$) whose focus lies on the vertex of the 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 points on the parabola $C_2$, where $y_1 \neq y_2$ and $y_1 + y_2 = 4$. The perpendicular bisector of 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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