
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 $f(x) = -\sqrt{3}\sin x \sin\left(x + \frac{\pi}{2}\right) + \cos^2 x - \frac{1}{2}$ ($x \in \mathbb{R}$).
1. Find the interval of monotonic increase for the function $f(x)$.
2. The graph of the function $f(x)$ is transformed by doubling all the x-coordinates of the points on the graph and then shifting it to the right by $\frac{\pi}{6}$ units to obtain the graph of $g(x)$. Find the maximum and minimum values of the function $y = g(x)$ for $x \in [0, \pi]$.

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