
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
Points \( E, F, G, H \) are the midpoints of edges \( AB, A_{1}D_{1}, A_{1}B_{1}, BC \) of a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with edge length \( a \). Determine the ratio of the circumradius to the inradius of the tetrahedron \( EFGH \).

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