
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
74. \(ABCD\) is a regular tetrahedron with edge length \(a\). Let \(M\) be the center of face \(ADC\), and \(N\) be the midpoint of edge \(BC\). Find the radius of the sphere inscribed in the trihedral angle at vertex \(A\) and tangent to the line \(MN\).

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