
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 $P$ is a point on the ellipse, with $2|F_1F_2| = |PF_1| + |PF_2|$.  
(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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