
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]


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Given that a hyperbola with an eccentricity of $e$ and an ellipse with an eccentricity of $\frac{\sqrt{2}}{2}$ share the same foci $F\_1$, $F\_2$, and $P$ is a common point on both curves with $\angle F\_1 P F\_2 = \frac{\pi}{3}$, determine the value of $e$.

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