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Let's put into the black box whole set $S=\{x_i\}_{i=1}^n$. If $bb(S)$ is True, then such a subset exists and we can go on:

1. R:=S
2. for i:=1 to n do
1. If $bb(R/\{x_i\})$ is True then $R:=R/\{x_i\}$

When this iteration is finished R will be subset of S that adds up to k.

This above solution assumes that there is only a single subset of S that adds to k.