
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 real numbers \( x \) in the interval \( 0 \leqslant x \leqslant 2 \pi \) that satisfy the inequality 
\[ 
2 \cos x \leqslant|\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}| \leqslant \sqrt{2}.
\]

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