
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
[IMO 2007 HKTST 1](http://www.mathlinks.ro/Forum/viewtopic.php?t=107262)
Problem 6

Determine all pairs  $(x,y)$  of positive integers such that  $\frac{x^{2}y+x+y}{xy^{2}+y+11}$  is an integer.

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