
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
Let point P be an intersection of an ellipse M and a hyperbola Γ, both having common foci F₁ and F₂. Given that $\cos\angle F_{1}PF_{2} = \frac{4}{5}$, if the eccentricity of ellipse M is $e_1$ and the eccentricity of hyperbola Γ is $e_2$, with $e_2 = 2e_1$, find the value of $e_1$.

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