
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 $x$ and $y$ be real numbers such that $x \neq y$ and $xy = 1$. Find $x+y$ given that
\[
\begin{vmatrix} 1 & 5 & 8 \\ 3 & x & y \\ 3 & y & x \end{vmatrix} = 0.
\]

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