
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
Let the function $f(x) = 4\sin x \cdot \sin^2\left(\frac{\pi + 2x}{4}\right) - \sin^2 x + \cos^2 x$.

(1) Find the intervals of monotonic increase for the function $f(x)$ in $[0, 2\pi)$;  
(2) Let set $A = \{x | \frac{\pi}{6} \leq x \leq \frac{2\pi}{3}\}$, and $B = \{x | -2 < f(x) - m < 2\}$. If $A \subseteq B$, find the range of the real number $m$.

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