
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 $f(x)=\sin \omega x\cos \varphi +\cos \omega x\sin \varphi$, where $\omega > 0$ and $|\varphi| < \frac{π}{2}$.

$(Ⅰ)$ If $f(0)=-\frac{{\sqrt{3}}}{2}$, find the value of $\varphi$.

$(Ⅱ)$ If $f(x)$ is monotonically increasing on $[-\frac{π}{3}$, $\frac{{2π}}{3}]$ and $f(\frac{{2π}}{3})=1$, choose one of the following conditions as known and find the values of $\omega$ and $\varphi$.

Condition 1: $f(\frac{π}{3})=1$;

Condition 2: $f(-\frac{π}{3})=-1$;

Condition 3: $f(x)$ is monotonically decreasing on $[-\frac{{4π}}{3}$, $-\frac{π}{3}]$.

Note: If multiple conditions are chosen and answered separately, the first answer will be scored.

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