One$-$Third$-$Subset$-$Sum: Given the same inputs, determine if there exists a subset S$\{$a$1,$a$2,...,$an$\}$S $\backslash$subseteq $\backslash \{$a$\_1,$ a$\_2,\backslash$dots$,$ a$\_$n$\backslash \}$S$\{$a$1,$a$2,...,$an$\}$ such that:

The cardinality of SSS$,$S$|$S$|$S$,$ is at most $$n$/3\backslash$lfloor n $/3\backslash$rfloor$$n$/3,$The sum of the elements in SSS equals BBB$,$ i$.$e$.,$a in Sa$=$B$\backslash$sum$\_\{$a $\backslash$in S$\}$ a $=$ B$$a in S$$a$=$B$.$

Goal:

Prove that if One$-$Third$-$Subset$-$Sum has a polynomial$-$time algorithm, then Subset$-$Sum also has a polynomial$-$time algorithm.