
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 $f(x)=A\sin(\omega x+\varphi)$ $(A \gt 0$, $\omega \gt 0$, $0 \lt \varphi \lt \frac{π}{2})$, choose three out of the following four conditions:<br/>① The maximum value is $2$;<br/>② The minimum positive period is $2\pi$;<br/>③ $f(0)=-2$;<br/>④ $f(-\frac{π}{6})=0$.<br/>$(Ⅰ)$ Find the analytical expression of the function $f(x)$ and the interval of monotonic decrease;<br/>$(Ⅱ)$ Let $g(x)=f(x)f(x-\frac{π}{6})$. When $x\in [0,m]$, the range of $g(x)$ is $[0,2+\sqrt{3}]$. Find the range of $m$.

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