
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 $\triangle ABC$, the angles $A$, $B$, and $C$ are the three internal angles, and $B = 2A$. The vectors $\overrightarrow{m} = (\cos A, -\sin B)$ and $\overrightarrow{n} = (\cos B, \sin A)$ are orthogonal to each other.
(1) Find the size of angle $B$.
(2) Let $f(x) = \cos (\omega x - \frac{B}{2}) + \sin \omega x$ $(\omega > 0)$, and the smallest positive period of $f(x)$ is $\pi$. Find the monotonically increasing interval of $f(x)$ and the maximum value of $f(x)$ on $[0, \frac{\pi}{2}]$.

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