
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 $f(x)=\cos (4x- \frac {\pi}{3})+2\cos ^{2}(2x)$, if we stretch the x-coordinates of all points on the graph of $y=f(x)$ by a factor of $2$ without changing the y-coordinates, and then shift the resulting graph to the right by $\frac {\pi}{6}$ units to obtain the graph of $y=g(x)$, determine an interval where the function $y=g(x)$ is increasing.

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