
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
In an acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Define vectors $\overrightarrow {m} = (2\sin B, \sqrt{3})$ and $\overrightarrow {n} = (2\cos^2 \frac{B}{2} - 1, \cos 2B)$, and it is given that $\overrightarrow {m}$ is perpendicular to $\overrightarrow {n}$.
(1) Find the magnitude of angle $B$.
(2) Find the intervals of monotonic increase for the function $f(x) = \sin 2x \cos B - \cos 2x \sin B$.
(3) If $b=4$, determine the range of possible areas for $\triangle ABC$.

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