
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
Determine all such pairs pf  positive integers  $(a, b)$  such that   $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot  lcm(a -1, b)$ , where  $lcm (a, b)$  denotes the smallest common multiple, and  $gcd (a, b)$  denotes the greatest common divisor of numbers  $a, b$ .

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