Let $S(A,B,C,D)$ be a relation schema and $F=-\{ABC,BDA\}$ be a set of functional dependencies on S$.$ Let S$1($A$,$B$,$C$)$ and S$2($C$,$D$)$ constitute a decomposition of S$.$

Find all keys of S$.$

Determine whether $S$ is in $BCNF,3NF$ or $2NF.$

Is the decomposition of S into S$1$ and S$2$ a lossless join decomposition?

What is the highest normal form of S$1$ and S$2.$