
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 circle $O\_1$: $(x-2)^2+y^2=16$ and circle $O\_2$: $x^2+y^2=r^2(0 < r < 2)$, find the minimum value of $e\_1+2e\_2$ where $e\_1$ and $e\_2$ are the eccentricities of the two ellipses formed by the trajectory of the moving circle's center $M$ as it is tangent to both $O\_1$ and $O\_2$.

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