
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  $A$  be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block  $a_1, a_2, \ldots, a_r$  becomes  $a_r, a_{r-1}, \ldots, a_1$ ).

As an example, let  $A$  be the sequence  $1,1,0,0,1$ . We can pick block  $1,0,0$  and flip it, so the sequence  $1,\boxed{1,0,0},1$  becomes  $1,\boxed{0,0,1},1$ . However, we cannot pick block  $1,1,0,0$  and flip their order since they contain the same number of  $1$ s and  $0$ s.

Two sequences  $A$  and  $B$  are called *related* if  $A$  can be transformed into  $B$  using a finite number the operation mentioned above.

Determine the largest natural number  $n$  for which there exists  $n$  different sequences  $A_1, A_2, \ldots, A_n$  where each sequence consists of 2022 digits, and for every index  $i \neq j$ , the sequence  $A_i$  is not related to  $A_j$ .

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