
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 complex number $z_1$ satisfies $| \sqrt{3}+i |=z_1-2i$

$(1)$ Find $z_1$

$(2)$ If the complex number $z_2$ satisfies $z_2-z_1=m^2-2m-5+(m-2)i$, and the point corresponding to $z_2$ in the complex plane is located in the first quadrant, find the range of values for $m$

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