
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 that the focus of ellipse $C$ lies on the $y$-axis, the length of its major axis is $4$, and the eccentricity is $\frac{\sqrt{3}}{2}$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ If point $P(0,-3)$, and the line $l:y=kx+1$ intersects ellipse $C$ at points $M$ and $N$, find the maximum area of triangle $PMN$.

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