
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
Transform the left-hand side of the equation \((\sin x + \cos y)(\cos x - \sin y)\):

\[
(\sin x + \cos y)(\cos x - \sin y) = \sin x \cos x + \cos y \cos x - \sin x \sin y - \cos y \sin y = \cos (x + y) + \frac{1}{2}(\sin 2x - \sin 2y) = \cos (x + y) + \sin (x - y) \cos (x + y)
\]

Substitute the obtained expression into the original equation:

\[
\cos (x + y) + \sin (x - y) \cos (x + y) = 1 + \sin (x - y) \cos (x + y)
\]

From here, we get \(\cos (x + y) = 1\). Solving this equation, we find:

\[
x + y = 2 \pi k, \; k \in \mathbb{Z}
\]

Therefore, the smallest positive value for \(x + y\) is \(2 \pi\).

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