
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  $S$  be a string of  $99$  characters,  $66$  of which are  $A$  and  $33$  are  $B$ . We call  $S$  *good* if, for each  $n$  such that  $1\le n \le 99$ , the sub-string made from the first  $n$  characters of  $S$  has an odd number of distinct permutations. How many good strings are there? Which strings are good?

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