
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
Four copies of an acute scalene triangle  $\mathcal T$ , one of whose sides has length  $3$ , are joined to form a tetrahedron with volume  $4$  and surface area  $24$ .  Compute the largest possible value for the circumradius of  $\mathcal T$ .

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