
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 $f(x) = \frac{\sqrt{3}}{3} \cos 2x + \sin^2(x + \frac{\pi}{4})$,

(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;

(2) Find the range of $f(x)$ when $x \in [-\frac{\pi}{12}, \frac{5\pi}{12}]$.

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