
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
Given the function $f(x) = 2\sin x\cos x + \sin(2x + \frac{\pi}{2})$.
(1) If $x \in \mathbb{R}$, find the smallest positive period and the intervals of monotonic increase for $f(x)$;
(2) When $x \in [0, \frac{\pi}{3}]$, find the maximum value of $f(x)$.

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