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For an integer $n\ge 3$, we introduce a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, referred to as one-sided $n$-suspended categories. Notably, one-sided $n$-angulated categories are specific instances of this structure. We establish a framework for transitioning from these generalized categories to their $n$-angulated counterparts. Additionally, we present a method for constructing $n$-angulated quotient categories from Frobenius $n$-prile categories. Our results unify and extend...