
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 unit cube $A B C D - A_{1} B_{1} C_{1} D_{1}$ with the midpoints of its edges $A B$, $A_{1} D_{1}$, $A_{1} B_{1}$, and $B C$ being $L$, $M$, $N$, and $K$ respectively, find the radius of the inscribed sphere of the tetrahedron $L M N K$.

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