
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 a point $P$ that is an intersection of an ellipse $M$ and a hyperbola $\Gamma$, both sharing common foci $F\_1$ and $F\_2$. If $\cos \angle F\_1 P F\_2 = \frac{4}{5}$, the eccentricity of ellipse $M$ is $e\_1$, and the eccentricity of hyperbola $\Gamma$ is $e\_2$. If $e\_2 = 2e\_1$, find the value of $e\_1$.

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