
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  $a$  and  $b$  be positive integers. As is known, the division of of  $a \cdot  b$  with  $a + b$  determines  integers  $q$  and  $r$  uniquely such that  $a \cdot b = q (a + b) + r$  and  $0 \le  r <a + b$ . Find all pairs  $(a, b)$  for which  $q^2 + r = 2011$ .

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