
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
Let  $n$  be a natural number. Find the least natural number  $k$  for which there exist  $k$  sequences of  $0$  and  $1$  of length  $2n+2$  with the following property: any sequence of  $0$  and  $1$  of length  $2n+2$  coincides with some of these  $k$  sequences in at least  $n+2$  positions.

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