
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
Given that $x$ and $y$ are positive integers, and satisfy $xy - (x + y) = 2p + q$, where $p$ and $q$ are the greatest common divisor and the least common multiple of $x$ and $y$, respectively. Find all such pairs of $(x, y)$ ($x \geq y$).

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