
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 $f(x)=-2a\sin(2x+\frac{\pi}{6})+2a+b$,
(1) If $a=1$ and $b=-1$, find the maximum and minimum values of $f(x)$;
(2) When $x \in [\frac{\pi}{4}, \frac{3\pi}{4}]$, does there exist constants $a$ and $b \in \mathbb{Q}$ such that the range of $f(x)$ is $[-3, \sqrt{3}-1]$? If so, find the values of $a$ and $b$; if not, explain why.

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