
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
Find all numbers \(x\) in the interval \(0 \leq x \leq 2\pi\) that satisfy the following inequalities:

\[ 
2 \cos x \leq |\sqrt{1+\sin 2x} - \sqrt{1-\sin 2x}| \leq \sqrt{2} 
\]

(International Mathematical Olympiad problem, 1965. CMO6511)

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