Give the answer of given question and also give details.
Question: Every company makes particular restrictions for passwords, but employees of ABC have particularly strange restrictions. Their passwords may only consist of the lowercase letters a, b and c. Furthermore, for any a in a password, it must be followed by a b or c. For any b in a password, it must be followed by an a, and c may be followed by any of the three letters.
For examples, abaccba is a valid password of length 7. (Note that for both b's, a comes right after them.) Using a transition matrix, find the number of valid passwords of length 12.
You may either use a computer program or a calculator to do the suitable matrix exponentiation and multiplication.
Provide answer in detail: no word count.