
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
A set  $S$  of integers is Balearic, if there are two (not necessarily distinct) elements $s,s'\in S$  whose sum  $s+s'$  is a power of two; otherwise it is called a non-Balearic set.
Find an integer  $n$  such that  $\{1,2,\ldots,n\}$  contains a 99-element non-Balearic set,
whereas all the 100-element subsets are Balearic.

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