
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
Isabel wants to partition the set  $\mathbb{N}$  of the positive integers into  $n$  disjoint sets  $A_{1}, A_{2}, \ldots, A_{n}$ . Suppose that for each  $i$  with  $1\leq i\leq n$ , given any positive integers  $r, s\in A_{i}$  with  $r\neq s$ , we have  $r+s\in A_{i}$ . If  $|A_{j}|=1$  for some  $j$ , find the greatest positive integer that may belong to  $A_{j}$ .

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