Searching for a Password

You are trying to recover a password to an encrypted file, by using search. You know that the password is up to $10$ letters long and contains only the letters A$,$ B$,$ C$.$ You formulate a search problem:

$$ The initial state is the empty string.

$$ The successor function is to append one letter $($A$,$ B$,$ or C$)$ to the string.

$$ The goal test is to verify a candidate password using the decryption software. There are $6$ correct passwords: AAACCC, ABBCC, BABAB, BCABACB, CBAC, and CBACB.

$$ Assume in the equal situation, visit would be alphabetically, i$.$e$.,$ expand $$A$$ first, then $$B$,$ then $$C$.$