
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
$(1)$ In the complex plane, if the complex numbers corresponding to $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are $7+i$ and $3-2i$ respectively, find $|\overrightarrow{AB}|$;<br/>$(2)$ The complex number $z$ satisfies $({1+2i})\overline{z}=4+3i$, find $z$;<br/>$(3)$ Given $m\in R$, the complex number $z=\frac{{m({m-2})}}{{m-1}}+({{m^2}+2m-3})i$, for what values of $m$,<br/>①$z\in R$; ②$z$ is a purely imaginary number.

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