
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 $f(x) = \sqrt{3}\sin(x)\cos(x) - \cos^2(x) + \frac{1}{2}$ with $x \in \mathbb{R}$,
(1) Determine the intervals on which $f(x)$ is monotonically increasing;
(2) If all the x-coordinates of the points on the graph of $f(x)$ are doubled and then the graph is shifted right by $\frac{\pi}{6}$ units, we obtain the graph of $g(x)$. Find the maximum and minimum values of $y = g(x)$ for $x \in [0,\pi]$.

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