
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 $\overrightarrow{m}=(2\cos \omega x,1)$ and $\overrightarrow{n}=(\sqrt{3}\sin \omega x-\cos \omega x,a)$, where $(x\in\mathbb{R},\omega > 0)$, the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ has a minimum positive period of $\pi$ and a maximum value of $3$.

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

(Ⅱ) Find the intervals of monotonic increase for the function $f(x)$.

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