Consider the following set F of functional dependencies on the relation schema R$($A$,$ B$,$ C$,$ D$,$ E$,$ F$)$:

A$->$ D

AB$->$ DE

BD$->$ A

AC$->$ F

$($a$)$ List two candidate keys for R$.$

$($b$)$ Give a lossless dependency preserving $3$NF decomposition of r based on the set of functional dependencies.

$($c$)$ Give a lossless BCNF decomposition of r based on the set of functional dependencies. Does the decomposition preserve dependencies? Why or why not?