
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 equation $x^{2}-px+25=0$ with roots $x_{1}$ and $x_{2}$ in the complex number range, where $p\in \mathbb{R}$.
$(1)$ If $p=8$, find $x_{1}$ and $x_{2}$;
$(2)$ If $x_{1}=3+4i$, find the value of $p$.

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