
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
Solve the equation \(\cos 2x + \cos 6x + 2 \sin^2 x = 1\).

In your answer, specify the number equal to the sum of the equation's roots that belong to the segment \(A\), rounding this number to two decimal places if necessary.

$$
A=\left[\frac{5 \pi}{6} ; \frac{6 \pi}{6}\right]
$$

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