
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  $S$  consists of  $k$  sequences of  $0,1,2$  of length  $n$ . For any two sequences  $(a_i),(b_i)\in S$  we can construct a new sequence  $(c_i)$  such that  $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$  and include it in  $S$ . Assume that after performing finitely many such operations we obtain all the  $3n$  sequences of  $0,1,2$  of length  $n$ . Find the least possible value of  $k$ .

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