
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 a focal length of $2\sqrt{2}$ and equation $\frac{x^2}{a} + y^2 = 1$ $(a > 1)$, let the two foci be $F_1$ and $F_2$, and let point $P$ lie on the ellipse. If $|PF_1| = 2$, then find the value of $|PF_2|$.

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