Recursive decomposition

$($a$)$ A binary string is a palindrome if it reads the same forwards and backwards. Examples of palindromes are $01100110,01010,\backslash$epsi $.$

Let P be the set of all binary strings that are palindromes. Give a recursive decomposition for P$,$ with a brief explanation. Then use it to determine the generating series for P with respect to length.

$($b$)$ Let A be the set of all binary strings that do not contain $0110011001$ as a substring.

Let B be the set of all binary strings with exactly one copy of $0110011001,$ at the right end. Write down two equations relating A and B with a brief explanation. Then use them to determine the generating series for A with respect to length.