
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 center of ellipse $C$ is at the origin, the foci are on the $x$-axis, the length of the major axis is $4$, and the point $({1\;,\;\frac{{\sqrt{3}}}{2}})$ lies on ellipse $C$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ If point $P$ is in the second quadrant and $\angle F_{2}PF_{1}=60^{\circ}$, find the area of $\triangle PF_{1}F_{2}$.

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