
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 an ellipse with two foci $F_{1}(-1,0)$ and $F_{2}(1,0)$, and $P$ is a point on the ellipse. It is known that $2|F_{1}F_{2}| = |PF_{1}| + |PF_{2}|$.  
(1) Find the equation of this ellipse;  
(2) If point $P$ is in the second quadrant and $\angle F_{2}F_{1}P = 120^{\circ}$, find the area of the triangle $\triangle PF_{1}F_{2}$.

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