
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 an ellipse are $F_{1}(-1,0)$, $F_{2}(1,0)$, and $P$ is a point on the ellipse, and $|F_{1}F_{2}|$ is the arithmetic mean of $|PF_{1}|$ and $|PF_{2}|$, derive the equation of the ellipse.

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