
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 that $f(x)=2A\cos^2(\omega x+\varphi)-A$ with constraints $A > 0$, $\omega > 0$, and $0 < \varphi < \frac{\pi}{2}$, the line $x= \frac{\pi}{3}$ and the point $\left(\frac{\pi}{12},0\right)$ are respectively a symmetry axis and a symmetry center on the graph of $f(x)$. Determine the interval(s) where the function $f(x)$ is increasing.

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