
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 vectors $\overset{\rightharpoonup}{m} = \left(2\cos \omega x,\ 1\right)$, $\overset{\rightharpoonup}{n} = \left( \sqrt{3}\sin \omega x - \cos \omega x, a \right)$, where $(x \in \mathbb{R}, \omega > 0)$, the smallest positive period of the function $f(x) = \overset{\rightharpoonup}{m} \cdot \overset{\rightharpoonup}{n}$ is $\pi$, and its maximum value is $3$.

(I) Find the values of $\omega$ and constant $a$;

(II) Find the interval(s) where function $f(x)$ is monotonically increasing.

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