
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 $f(x)=\sin\left(2x+\frac{\pi}{6}\right)$, if the graph is shifted to the right by $\frac{\pi}{6}$ units, and then all the x-coordinates of the resulting graph are changed to $\frac{1}{\omega}$ times their original values (where $\omega>0$), while the y-coordinates remain unchanged, find the range of values for $\omega$, given that the resulting function $g(x)$ has exactly $2$ zeros in the interval $[0,\frac{\pi}{4}]$.

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