
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
Let $a$ be a real number, and let complex numbers $z_{1}=1+2i$ and $z_{2}=(a+i)(1-i)(i)$ (where $i$ is the imaginary unit).<br/>$(1)$ If the point corresponding to $z_{2}$ in the complex plane is in the first quadrant, find the range of values for $a$;<br/>$(2)$ Find the minimum value of $|\overline{{z}_{1}}+z_{2}|$.

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