
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  $k$  be a positive integer. Lexi has a dictionary  $\mathbb{D}$  consisting of some  $k$ -letter strings containing only the letters  $A$  and  $B$ . Lexi would like to write either the letter  $A$  or the letter  $B$  in each cell of a  $k \times k$  grid so that each column contains a string from  $\mathbb{D}$  when read from top-to-bottom and each row contains a string from  $\mathbb{D}$  when read from left-to-right. 
What is the smallest integer  $m$  such that if  $\mathbb{D}$  contains at least  $m$  different strings, then Lexi can fill her grid in this manner, no matter what strings are in  $\mathbb{D}$ ?

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