
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
It is known that the ellipse $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) and the hyperbola $C_2: x^2 - y^2 = 4$ share the same right focus $F_2$. Point $P$ is a common point of $C_1$ and $C_2$. If $|PF_2| = 2$, then the eccentricity of the ellipse $C_1$ is ______.

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