
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
Let the complex number $z=a+bi$ $(a,b\in \mathbb{R}, a > 0)$ satisfy $|z|= \sqrt{10}$, and the complex number $(1-2i)z$ corresponds to a point on the bisector of the second and fourth quadrants in the complex plane.

$(1)$ Find the complex number $Z$;  
$(2)$ If $\overline{2}+ \frac{m+i}{1-i}$ $(m\in \mathbb{R})$ is a pure imaginary number, find the value of the real number $m$.

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