
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 vectors $\overrightarrow{a} = (2\cos x, 1)$ and $\overrightarrow{b} = (\cos x, \sqrt{3}\sin 2x + m)$, let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 1$.  
(1) Find the intervals of increase for $f(x)$ when $x \in [0, \pi]$.  
(2) When $x \in [0, \frac{\pi}{6}]$, if $-4 \leq f(x) \leq 4$ always holds, find the range of the real number $m$.

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