
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 complex numbers $z_{1}=a-5i$ and $z_{2}=-2+bi\left(a,b\in R\right)$ are conjugate to each other.<br/>$(1)$ Determine in which quadrant the point corresponding to $z_{2}$ lies in the complex plane, and explain the reason;<br/>$(2)$ Solve the quadratic equation in terms of $z$: $z^{2}+az+b=0$ in the complex number field.

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