
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
Given the natural  $n$ . We shall call *word* sequence from  $n$  letters of the alphabet, and *distance*  $\rho(A, B)$   between *words*  $A=a_1a_2\dots a_n$  and  $B=b_1b_2\dots b_n$  , the number of digits in which they differ (that is, the number of such  $i$ , for which  $a_i\ne b_i$ ). We will say that the *word*  $C$   *lies* between words   $A$  and    $B$  , if  $\rho (A,B)=\rho(A,C)+\rho(C,B)$ . What is the largest number of *words* you can choose so that among any three, there is a *word lying* between the other two?

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