$4.(20$pt$)$ Prove that the following problems are decidable: $1($a$)(10$pt$)$ Given a regular language L over $\{$a$,$ b$\},$ is it true that L contains infinitely many strings that have at least one a but finitely many strings that have at most one b$?($b$)(10$pt$)$ Given two regular languages, L$1$ and L$2,$ is it true that every string in L$1$ is a prefix of a concatenation of strings from L$2?$