
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 triangular prism with an equilateral base of side length $m$. The lateral edges are equal to the side length of the base. One of the vertices is equidistant from the vertices of the base. Find the largest radius of the sphere that can fit inside the prism.

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