
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
A student used the "five-point method" to draw the graph of the function $f\left(x\right)=A\sin \left(\omega x+\varphi \right)$ $(A \gt 0$, $\omega\ \ \gt 0$, $|\varphi | \lt \frac{π}{2})$ within one period. The student listed and filled in some of the data in the table below:

| $\omega x+\varphi $ | $0$ | $\frac{π}{2}$ | $\pi $ | $\frac{3π}{2}$ | $2\pi $ |
|---------------------|-----|---------------|--------|-----------------|--------|
| $x$                 |     | $\frac{π}{6}$ |        | $\frac{2π}{3}$  |        |
| $y=A\sin \left(\omega x+\varphi \right)$ | $0$ | $2$ | $0$ |     | $0$ |

$(Ⅰ)$ The analytical expression of the function $f\left(x\right)$ is $f\left(x\right)=\_\_\_\_\_\_$ (write down the result directly);<br/>
$(Ⅱ)$ Find the intervals where the function $f\left(x\right)$ is monotonically increasing;<br/>
$(Ⅲ)$ Find the minimum value of the function $f\left(x\right)$ on the interval $[-\frac{π}{2}$, $0]$.

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