
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 $f(x)=A\sin(\omega x+\varphi)$, where $A>0$, $\omega>0$, and $|\varphi|\leqslant \pi$, has a minimum value of $-3$ when $x=\dfrac{\pi}{6}$ within one period, and a maximum value of $3$ when $x=\dfrac{2\pi}{3}$.

(Ⅰ) Find the analytical expression of $f(x)$.

(Ⅱ) Find the extreme values of $f(x)$ on the interval $[\dfrac{\pi}{2}, \pi]$.

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