
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 the complex numbers $z=\frac{3+i}{2-i}$, $z_{1}=2+mi$.

$(1)$ If $|z+z_{1}|=5$, find the value of the real number $m$;

$(2)$ If the point corresponding to the complex number $az+2i$ on the complex plane is in the second quadrant, find the range of values for the real number $a$.

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