
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
Let the function $f(x) = \vec{a} \cdot \vec{b}$, where vector $\vec{a} = (2\cos x, 1)$, and $\vec{b} = (\cos x, \sqrt{3}\sin 2x + m)$.  
$(1)$ Find the smallest positive period of the function $f(x)$ and the interval of monotonic increase on $[0, \pi]$.  
$(2)$ When $x \in [0, \frac{\pi}{6}]$, the maximum value of $f(x)$ is $4$, find the value of $m$.

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