
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
Consider all binary sequences of length  $n$ . In a sequence that allows the interchange of positions of an arbitrary set of  $k$  adjacent numbers, ( $k < n$ ), two sequences are said to be *equivalent* if they can be transformed from one sequence to another by a finite number of transitions as above. Find the number of sequences that are not equivalent.

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