
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
The function $f(x)=\sin(\omega x+\frac{\pi}{3})+\cos(\omega x-\frac{\pi}{6})(\omega>0)$, when all the abscissas of the points on the graph of $f(x)$ are shortened to half of their original length (while the ordinates remain unchanged), the graph of the function $g(x)$ is obtained. If $g(x)$ has exactly one extremum point in the interval $(0,\frac{\pi}{18})$, then the smallest integer value of $\omega$ is ______.

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