
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 function $f(x)=2\cos x\sin \left(x+ \frac {\pi}{6}\right)+1$, $x\in\mathbb{R}$.  
(I) Find the smallest positive period of the function $f(x)$ and the intervals where it is monotonically increasing;  
(II) If the graph of $f(x)$ is translated to the right by $\frac {\pi}{3}$ units to obtain the graph of the function $g(x)$, and $x\in\left[- \frac {\pi}{6}, \frac {\pi}{3}\right]$, find the range of the function $g(x)$.

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