
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 $z$ be non-zero numbers such that $y = \frac{1}{x} + z$. Evaluate the expression $\left(x - \frac{1}{x}\right)\left(y + \frac{1}{y}\right)$.
**A)** $\frac{x^4 - 1}{x^2(1+xz)}$  
**B)** $\frac{(x^2 - 1)(1 + 2xz + x^2z^2 + x^2)}{x^2(1 + xz)}$  
**C)** $\frac{x^2 - 1}{x^2(1+xz)}$  
**D)** $\frac{x^4 + x^2 - 1}{x^2(1 + xz)}$

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