
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
By adding the first two equations and subtracting the third, we obtain:

\[
\cos x + \cos y + \cos z + \cos (x + y + z) = 0 \Leftrightarrow
\]

\[
\Leftrightarrow 2 \cos \left(\frac{x+y}{2}\right) \cdot \cos \left(\frac{x-y}{2}\right) + 2 \cos \left(\frac{z+x+y+z}{2}\right) \cdot \cos \left(\frac{y+x}{2}\right) = 0 \Leftrightarrow
\]

\[
\Leftrightarrow \cos \left(\frac{x+y}{2}\right) \cdot \left(\cos \left(\frac{x-y}{2}\right) + \cos \left(\left(\frac{x+y}{2}\right) + z \right) \right) = 0 \Leftrightarrow
\]

\[
\Leftrightarrow \cos \left(\frac{x+y}{2}\right) \cdot \cos \left(\frac{x+z}{2}\right) \cdot \cos \left(\frac{y+z}{2}\right) = 0 \Leftrightarrow
\]

\[
\left[\begin{array}{c}
\cos \left(\frac{x+y}{2}\right) = 0 \\
\cos \left(\frac{x+z}{2}\right) = 0 \\
\cos \left(\frac{y+z}{2}\right) = 0
\end{array}\right.
\]

\[
\left[\begin{array}{c}
x + y = \pi + 2\pi n \\
x + z = \pi + 2\pi m \\
y + z = \pi + 2\pi k, \quad n, m, k \in \mathbb{Z}
\end{array}\right.
\]

Substituting \(x + y = \pi + 2\pi n\) into the system, we obtain:

\[
\left\{\begin{array}{l}
\cos x - 2 \cos z = 0 \\
\cos y - 3 \cos z = 0 \\
-5 \cos z = 0
\end{array}
\Leftrightarrow 
\left\{\begin{array}{l}
\cos x = 0 \\
\cos y = 0 \\
\cos z = 0
\end{array}
\Leftrightarrow
\left\{\begin{array}{l}
x = \frac{\pi}{2} + \pi n \\
y = \frac{\pi}{2} + \pi m \\
z = \frac{\pi}{2} + \pi k, \quad n, m, k \in \mathbb{Z}
\end{array}\right.
\right.
\right.
\]

The remaining cases are considered similarly.

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