
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 \( s_1 s_2 \ldots s_n \), we define its 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 \( B_n \).

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