
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 positive integer. Let  $B_n$  be the set of all binary strings of length  $n$ . For a binary string  $s_1\hdots s_n$ , we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by  $b$ . Then, we replace  $s_b$  with  $1-s_b$ . A string  $a$  is said to be a *descendant* of  $b$  if  $a$  can be obtained from  $b$  through a finite number of twists. A subset of  $B_n$  is called *divided* if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of  $B_n$ .

*Remark.* Here is an example of a twist:  $101100 \rightarrow 101000$  because  $1\mid 0\mid 11\mid 00$  has  $4$  blocks of consecutive digits. 

*Viktor Simjanoski*

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