
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 graph of the function $f(x)=\sin \omega x$ ($\omega > 0$) is shifted to the right by $\dfrac{\pi}{12}$ units to obtain the graph of the function $y=g(x)$. Additionally, the function $g(x)$ is monotonically increasing in the interval $\left[\dfrac{\pi}{6}, \dfrac{\pi}{3}\right]$ and monotonically decreasing in the interval $\left[\dfrac{\pi}{3}, \dfrac{\pi}{2}\right]$. Determine the value of the real number $\omega$.

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