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

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