
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
First, shift the graph of the function $f(x)=\sin x$ to the left by $\dfrac{\pi}{6}$ units, then change the x-coordinates of each point to $\dfrac{1}{\omega}$ times the original (where $\omega\in\mathbb{N}^{*}$) to obtain the graph of function $g(x)$. If $g(x)$ is monotonically increasing in the interval $\left[\dfrac{\pi}{6}, \dfrac{\pi}{4}\right]$, then the maximum value of $\omega$ is \_\_\_\_\_\_.

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