
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 vectors $\overrightarrow{m}=(2\cos x+2\sqrt{3}\sin x, 1)$ and $\overrightarrow{n}=(\cos x, -y)$ with $\overrightarrow{m} \perp \overrightarrow{n}$. Express $y$ as a function of $x$, denoted as $f(x)$.

1. Find the interval(s) where $f(x)$ is monotonically increasing.
2. Translate the graph of $f(x)$ to the right by $\frac{\pi}{6}$ units, and then double the value of each point's abscissa (x-coordinate) without changing the ordinate (y-coordinate) to obtain the graph of $g(x)$. Determine the maximum and minimum values of $g(x)$ on the interval $x \in [0,\pi]$.

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