
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
If $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, $\overrightarrow{a}=(4\sin \omega x,-1)$, $\overrightarrow{b}=(\cos(\omega x+\frac{\pi}{6}),-1)$, and the minimum distance from the center of symmetry of $f(x)$ to the axis of symmetry is $\frac{\pi}{8}$. <br/>$(1)$ Find the monotonic intervals of $f(x)$; <br/>$(2)$ Find the range of $f(x)$ on $[0,\frac{\pi}{3}]$.

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