Two compounds, A and $B,$ have the same molecular formula of $C_6{H}_{12}.$ Determining the structure of unknown organic molecules is a common, and often challenging, task. One tool we can use is to subject a mystery molecule to a series of chemical reactions and analye the products. In a well$-$designed analysis, only a single substrate can make all the observed products. Use this strategy to identify the structures of mystery molecules A and $B$ below.

a$)$ Reaction of A with $B{r}_2$ produces an achiral compound, $C,$ with molecular formula $C_6{H}_{12}B{r}_2.$ When one mole of $A$ is subjected to ozonolysis, $2$ moles of a single ketone product $($compound $D)$ are produced. Determine the structures of $A,C,$ and $D.$

b$)$ Reaction of $B$ with aqueous $C{l}_2$ produces an achiral chlorohydrin, $E.$ When $B$ is subjected to ozonolysis, the products are an aldehyde $($compound $F)$ and a ketone $($compound $G).$ Determine the structures of $B,E,$ and $F.$

Compound G $($a ketone$)$