
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 $\overrightarrow{a}=(3,-\cos (ωx))$ and $\overrightarrow{b}=(\sin (ωx), \sqrt {3})$, where $ω > 0$, the minimum positive period of the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$ is $π$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing.
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $f(\frac{A}{2}) = \sqrt{3}$, $a = \sqrt{3}b$. Find the measures of angles $A$, $B$, $C$.

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