let $\boldsymbol{X}_1,\,\dots,\,\boldsymbol{X}_n$ be well-ordered sets of positive values and $\mathcal{R}:=\lbrace\left(x_1,\,\dots,\,x_n\right)\rbrace = \boldsymbol{X}_1\times\,\dots\,\times\boldsymbol{X}_n$ the "induced" coordinate space.

Question:given $\boldsymbol{x}:=(x_1,\,\cdots,\,x_n)\in\mathcal{R}$, how can one determine $$\boldsymbol{y}:=(y_1,\,\cdots,\,y_n)\in\mathcal{R}:\\\quad \|\boldsymbol{x}\|_1\,\lt\,\|\boldsymbol{y}\|_1\quad \land\quad \|\boldsymbol{x}\|_1\,\lt\,\|\boldsymbol{z}\|_1\implies \|\boldsymbol{y}\|_1\,\le\,\|\boldsymbol{z}\|_1$$ under the assumption that all norms are different and the elements of the $\boldsymbol{X}_i$ are sorted in ascending order?

As I intend to utilize the successor-generation in a branch and bound algorithm, I am looking for algorithms with minimal memory footprint.

Disclaimer: I already have an idea for such an algorithm, but can't prove its correctness and would like to hear of existing solutions before sharing my idea.