
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

For all triples \((x, y, z)\) satisfying the system

\[
\left\{\begin{array}{l}
\sqrt{3} \sin x = \tan y \\
2 \sin y = \cot z \\
\sin z = 2 \tan x
\end{array}\right.
\]

find the minimum value of \(\cos x - \cos z\).

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