
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  $n$  be a positive integer. Let  $S$  be a set of ordered pairs  $(x, y)$  such that  $1\leq x \leq n$  and  $0 \leq y \leq n$  in each pair, and there are no pairs  $(a, b)$  and  $(c, d)$  of different elements in  $S$  such that  $a^2+b^2$  divides both $ac+bd$  and  $ad - bc$ . In terms of  $n$ , determine the size of the largest possible set  $S$ .

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