
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 a complex number $z=m+(4-m^{2})i$ (where $m$ is a positive real number) and $z+5i\in R$.  
$(1)$ Find $z$;  
$(2)$ If ${z}_{1}=\overline{z}(a+i)$ corresponds to a point in the second quadrant of the complex plane, find the range of real number $a$.

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