
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 $z$ such that $|z|=2\sqrt{10}$, and the complex number $\left(3-i\right)z$ is purely imaginary.  
$(1)$ Find $z$;  
$(2)$ If the real part of $z$ is negative, and $z$ is a root of the equation $2x^{2}+mx-n=0\left(m,n\in R\right)$, find the value of $m+n$.

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