
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 $f(x)=\sin (2\omega x- \frac {\pi}{6})+2\cos ^{2}\omega x-1$ ($\omega > 0$).  
$(1)$ If $\omega=1$, find the intervals of increase for the function $f(x)$.  
$(2)$ If the distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac {3\pi}{16}$, find the range of $f(x)$ on $\left[0, \frac {\pi}{8}\right]$.

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