
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
In the complex plane, the complex number corresponding to point $A$ is $\sqrt{3}+i$. The vector $\overrightarrow{OA}$ rotates counterclockwise around the origin $O$ by $120^\circ$ to obtain the vector $\overrightarrow{OC}$.  
$(1)$ Find the complex number $z_{0}$ corresponding to point $C$.  
$(2)$ Given that the complex number corresponding to point $B$ is $z$ and satisfies $|z-z_{0}|=1$, and $∠\overrightarrow{CB}$, $\overrightarrow{OC}=120°$, find the complex number $z$.

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