
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 ${z_1}={a^2}+2ai(a\in R)$, the vector corresponding to the complex number $z_{2}$ in the complex plane is $\overrightarrow{OA}=(-1,2)$.<br/>$(1)$ If $z_{1}+z_{2}$ is a purely imaginary number, find the value of $a$;<br/>$(2)$ If the point corresponding to $z_{1}i-z_{2}$ in the complex plane is in the fourth quadrant, find the range of values for $a$.

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