
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
The numbers \( x \) and \( y \) satisfy the equations \( \sin y + \sin x + \cos 3x = 0 \) and \( \sin 2y - \sin 2x = \cos 4x + \cos 2x \). What is the maximum value that the sum \( \cos y + \cos x \) can take?

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