$(20$pt$)$ Prove that the following problems are decidable:

$($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$)(10pt)$ 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 $L_2?$

You are allowed to use any of the following:

$-$ closure properties: union, concatenation, Kleene star, complement, intersection, difference

$-$ conversion algorithms between DFSM$,$ NDFSM$,$ regular expressions, and regular grammars

$-$ decision algorithms: membership, emptiness, finiteness, totality, equivalence, minimality.

Explain clearly which of the above closure property and algorithm you have used at each step.