
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
In the complex plane, the points corresponding to the complex numbers $z_{1}$ and $z_{2}$ are $\left(1,2\right)$ and $\left(-1,1\right)$, respectively.  
$(1)$ Find $|z_{1}-z_{2}|$;  
$(2)$ Given that the real part of the imaginary number $z$ is equal to the imaginary part of the complex number $z_{1}z_{2}$, and $z$ is a root of the equation $2x^{2}+ax+10=0$ with respect to $x$, find the value of the real number $a$ and $z$.

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