Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound

We address a question posed by Herzog, Kumashiro, and Stamate regarding the upper bound for the multiplicity of any far-flung Gorenstein numerical semigroup. We answer this question in the affirmative by presenting a method for constructing certain far-flung Gorenstein numerical semigroups with maximal reduced type. Furthermore, using this method, we consider a question raised by Herzog, Hibi, and Stamate. More precisely, for any integer $t\ge 5$, we construct a certain far-flung Gorenstein numerical semigroup with type $t$, which is a counterexample to the question.