
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 \(y\) are such that the following equalities hold:
\[
\cos y + \cos x = \sin 3x
\]
and
\[
\sin 2y - \sin 2x = \cos 4x - \cos 2x.
\]
What is the minimum value of the sum \(\sin y + \sin x\)?

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