$1.(2$ pts$)$ Design a context$-$free grammar $($CFG$)$ for the language L over the alphabet $=\{$a$,$b$\}$such that L consists of all strings where the number of a$\text{'}$s is twice the number of b$\text{'}$s$.$

$2.(8$ pts$)$ Consider the context$-$free grammar defined by:

S$$aSb$$bSa$$aS$$bS$$

I. Identify the variables, terminals, and start symbol of this grammar.

II$.$ Describe the language generated by this grammar.

III. Provide a derivation for the string "ababb".

IV$.$ Is the string "ababab" in the language? Justify your answer.