
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
Find the smallest positive integer  $N$  such that there are no different sets  $A, B$  that satisfy the following conditions. (Here,  $N$  is not a power of  $2$ . That is,  $N \neq 1, 2^1, 2^2, \dots$ .)



- $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$
- $|A| = |B| \geq 1$
- Sum of elements in  $A$  and sum of elements in  $B$  are equal.


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