Can cyclic shifts of ABAB and ABBA be equal?

Question

Do there exist binary strings $A$ and $B$ so that:

1. the word $ABAB$ is a cyclic shift of $ABBA$ and
2. the word $AB$ does not equal $BA$?

Here $AB$ represents the concatenation of $A$ and $B$.

Answer 1

No; note that any cyclic shift of $ABAB$ still has the property that it is some sequence repeated twice. If this is equal to $ABBA$, then the repeated sequence is equal to both $AB$ and $BA$, contradicting that these are different.