
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 complex numbers $z_{1}$ and $z_{2}$ are $\left(1,-3\right)$ and $\left(a,1\right)$, respectively, where $a\in \mathbb{R}$. Also, $\frac{{z_{2}}}{{z_{1}}}$ is a purely imaginary number.  
$(1)$ Find the value of $a$;  
$(2)$ If the conjugate of $z_{1}$, denoted as $\overline{{z_{1}}}$, is a root of the equation $x^{2}+px+q=0$, find the values of the real numbers $p$ and $q$.

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