
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 $y=\sin x (x\in R)$ is transformed such that the horizontal coordinates of all points on the graph are changed to $\frac{1}{2}$ of their original values (the vertical coordinates remain unchanged), and then the resulting graph is shifted to the right by $\frac{\pi}{6}$ units. Determine the interval(s) where $y=g(x)$ is monotonically increasing.

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