
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
It is known that \(\frac{\sin 3 x}{(2 \cos 2 x+1) \sin 2 y}=\frac{1}{5}+\cos ^{2}(x-2 y)\) and \(\frac{\cos 3 x}{(1-2 \cos 2 x) \cos 2 y}=\frac{4}{5}+\sin ^{2}(x-2 y)\). Find all possible values of the expression \(\cos (x-6 y)\), given that there are at least two.

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