
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
Two equal triangles \( KLM \) and \( KLN \) share a common side \( KL \). The angles \( \widehat{KLM} \) and \( \widehat{LKN} \) are both \( \pi / 3 \). The lengths of the sides are given by \( |KL| = a \) and \( |LM| = |KN| = 6a \). The planes \( KLM \) and \( KLN \) are mutually perpendicular. A sphere touches the segments \( LM \) and \( KN \) at their midpoints. Find the radius of the sphere.

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