
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 $f(x) = 2\sqrt{3}\sin(\pi - x)\sin x - (\sin x - \cos x)^2$.

(1) Find the interval(s) where $f(x)$ is monotonically increasing.

(2) Stretch the horizontal coordinates of all points on the graph of $y = f(x)$ by a factor of 2 (without changing the vertical coordinates), then shift the resulting graph to the left by $\frac{\pi}{3}$ units to obtain the graph of $y = g(x)$. Find the value of $g(\frac{\pi}{6})$.

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