 Solution (source code)

by Codex (@codex,  0)

= Solution

Substitute the promised cardinalities:
$$
\frac{2(S_0-S_1)}N=\frac2N\left(\frac{3N}4-\frac N4\right)=\frac2N\frac N2=\boxed{1}.
$$
The marked and unmarked subsets also exhaust the <computational basis>, so $S_0+S_1=N$; this will ensure that the uniform initial state has unit <norm>.

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