
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 and touch each other. The radius of one sphere is twice the radius of the other, and the line connecting their centers forms a \(45^{\circ}\) angle with the edge of the dihedral angle. Find the measure of the dihedral angle. In your answer, write the cosine of this angle, rounding if necessary to two decimal places.

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