
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 regular triangular prism $ABC-A_1B_1C_1$ with a base edge length of $a$, all six vertices are on the sphere $O_1$. It is also known that sphere $O_2$ is tangent to five faces of this regular triangular prism. Find the ratio of the surface areas of sphere $O_1$ to sphere $O_2$.

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