
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
In plane \\\(\alpha\\), it is known that \\(AB \perpendicular BC\\. Planes \\(\beta\\) and \\(\gamma\\) are constructed passing through lines \\(AB\\) and \\(BC\\) respectively, such that the acute dihedral angle \\(\alpha-AB-\beta\\) is \\(\dfrac{\pi}{3}\\) and the acute dihedral angle \\(\alpha-BC-\gamma\\) is \\(\dfrac{\pi}{3}\\). Find the cosine of the acute dihedral angle formed by planes \\(\beta\\) and \\(\gamma\\.

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