
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 two foci of an ellipse are $F_1(-1,0)$ and $F_2(1,0)$, and point $P$ is on the ellipse, satisfying $2|F_1F_2|=|PF_1|+|PF_2|$.

$(1)$ Find the equation of this ellipse;

$(2)$ If point $P$ is in the second quadrant and $\angle F_2F_1P=120^{\circ}$, find the area of $\triangle PF_1F_2$.

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