Equivalence of RE and FA $(40$ points, $10$ points each$)$

Design a Regular Expression for a regular language $L,$ that accepts the set of all strings that do not contain the substring $00.$

a$.$ Convert your regular expression from $($a$)$ to an NFA.

b$.$ Convert your NFA to DFA.

c$.$ Check if this is minimized DFA, if not, minimize your DFA.