
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

For every integer  $n	\ge 2$  let  $B_n$  denote the set of all binary  $n$ -nuples of zeroes and ones, and split  $B_n$  into equivalence classes by letting two  $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers  $n 	\ge 2$  for which  $B_n$  splits into an odd number of equivalence classes.

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