
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 $f(x)=A\sin(\omega x+\varphi)+B$ where $(A>0, \omega>0, |\varphi|<\frac{π}{2})$ and the corresponding values within one period are as shown in the table below:

|x|-\frac{π}{6}|$\frac{π}{3}$|$\frac{{5π}}{6}$|$\frac{{4π}}{3}$|$\frac{{11π}}{6}$|
|---|---|---|---|---|---|
|$f(x)$|-1|1|3|1|-1|

$(1)$ Based on the data provided in the table, find the analytical expression of the function $f(x)$;  
$(2)$ Based on the result of $(1)$, if the function $y=f(nx) (n>0)$ has a minimum positive period of $\frac{{2π}}{3}$, find the range of the function $y=f(nx)$ on the interval $[0,\frac{π}{3}]$.

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