
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 $p$, $q$, and $r$ be nonzero real numbers. Find the number of real roots of the equation
\[
\begin{vmatrix} x & p & -r \\ -p & x & q \\ r & -q & x \end{vmatrix} = 0.
\]

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