TxO Math Bowl 2024 - Individuals A - Problem 14

Let $N$ be the number of strings (words) of length $7$ such that each character is in the set $\{A,B,C,D\}$, and that given any four consecutive characters from the string, they are not all distinct. Find the remainder when $N$ is divided by $100$.

For example, $ABCBBDA$ is valid, as out of $ABCB$, $BCBB$, $CBBD$, and $BBDA$, each of them contain at least one duplicate letter, but $ABCDABB$ is invalid, because the letters in $BCDA$ are all distinct