
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 spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is three times greater than the radius of the other, and the line joining the centers of the spheres forms an angle of $60^\circ$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In the answer, provide the cosine of this angle, rounding to two decimal places if necessary.

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