
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
Consider the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin 2x + 2m$, where $x \in \mathbb{R}$ and $m \in \mathbb{R}$.
(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.
(II) If $f(x)$ has a minimum value of 0 when $0 \leq x \leq \frac{\pi}{4}$, find the value of the real number $m$.

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