
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
The function $f(x)=\sin^4x+2\sqrt{3}\sin x \cos x - \cos^4x$.  
$(1)$ Find the interval where the function $f(x)$ is decreasing on $[0,\pi]$.  
$(2)$ Shift the graph of $y=f(x)$ to the right by $\frac{\pi}{6}$ units, then shrink the horizontal axis by $\frac{1}{2}$ (vertical axis remains unchanged) to obtain the graph of $y=g(x)$. Find the range of $g(x)$ when $x\in[\frac{\pi}{6},\frac{\pi}{3}]$.

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