
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 $f(x) = \sin(\omega x + \varphi)$. where $\omega > 0$. If the smallest positive period of $f(x)$ is $\pi$, and $f(\frac{π}{2}) = f(\frac{2π}{3})$;  
$(1)$ Find the values of $\omega$ and $\varphi$;  
$(2)$ If $|\varphi| < \frac{π}{2}$, find the range of $f(x)$ on the interval $[-\frac{π}{3}, \frac{π}{6}]$.

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