
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
Let $f(x)=2\sqrt{3}\sin(\pi-x)\sin x-(\sin x-\cos x)^{2}$.  
(I) Determine the intervals on which $f(x)$ is increasing;  
(II) On the graph of $y=f(x)$, if every horizontal coordinate of the points is stretched to twice its original value (vertical coordinate remains unchanged) and then the resulting graph is translated to the left by $\frac{\pi}{3}$ units, we get the graph of the function $y=g(x)$. Find the value of $g\left(\frac{\pi}{6}\right)$.

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