
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 $\vec{a}=(2\cos x,1)$, $\vec{b}=(\cos x, \sqrt{3} \sin 2x)$.  
(Ⅰ) Find the smallest positive period of the function $f(x)$ and the interval of monotonic increase;  
(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[- \frac{\pi}{4}, \frac{\pi}{6} \right]$.

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