
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
The numbers \(x, y, p\) and \(q\) are all integers. \(x\) and \(y\) are variable and \(p\) and \(q\) are constant and positive. The four integers are related by the equation \(x y = p x + q y\). When \(y\) takes its maximum possible value, which expression is equal to \(y - x\)?

A) \(pq - 1\)
B) \((p-1)(q-1)\)
C) \((p+1)(q-1)\)
D) \((p-1)(q+1)\)
E) \((p+1)(q+1)\)

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