
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 that $2-i$ is a root of the equation $x^{2}-mx+n=0$ where $m,n\in \mathbb{R}$.  
$(1)$ Find the values of $m$ and $n$;  
$(2)$ If $z=a^{2}-na+m+(a-m)i$ is purely imaginary, find the real value of $a$.

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