
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
How many values can the expression

$$
z = uv + (uv)^{-1}
$$

take if

$$
\begin{gathered}
u + u^{-1} = x \\
v + v^{-1} = y
\end{gathered}
$$

where \( x \) and \( y \) are distinct numbers with absolute values not less than 2? Write an equation in which \( u \) and \( v \) do not appear, and whose roots are all the possible values of \( z \).

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