
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  $x$ ,  $y$  are positive integers with  $x(x+1)|y(y+1)$ , but neither  $x$  nor  $x+1$  divides either of  $y$  or  $y+1$ , and  $x^2 + y^2$  as small as possible, find  $x^2 + y^2$ .

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