Forcing upper $\mathrm{\Sigma }$-uniformization in the presence of lower $\mathrm{\Pi }$-reduction or uniformization

We present a method which allows the combination of forcing uniformization on the $\mathrm{\Pi }$- and the $\mathrm{\Sigma }$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where ${\mathrm{\Pi }}_3^1$-reduction holds, ${\mathrm{\Pi }}_3^1$-uniformization fails, yet ${\mathrm{\Sigma }}_n^1$ uniformization is true for $n\ge 4$. We also construct a universe where ${\mathrm{\Pi }}_3^1$-uniformization holds and for every $n\ge 4,$ ${\mathrm{\Sigma }}_4^1$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\text{ZFC})$.