
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 $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $x \in \mathbb{R}$) satisfies the conditions: $f(\frac{\pi}{6}) = 1$, $f(\frac{\pi}{3}) = 0$, and $f(x)$ is monotonic on $(\frac{\pi}{6}, \frac{\pi}{4})$. The number of possible values for $\omega$ is $\_\_\_\_\_\_.$

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