
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 foci of the ellipse are $F_1(-1,0)$ and $F_2(1,0)$, with an eccentricity of $e=\frac{1}{2}$  
$(1)$ Find the equation of the ellipse;  
$(2)$ Let point $P$ be a point on the ellipse, and $\angle F_1PF_2=60^\circ$, find the area of $\triangle F_1PF_2$.

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