
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 two circles $O\_1$: $(x-2)^2 + y^2 = 16$ and $O\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\_1$ and $O\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\_1$ and $e\_2 (e\_1 > e\_2)$. Find the minimum value of $e\_1 + 2e\_2$.

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