When is the Frobenius number of a numerical semigroup larger than the maximum of the minimal generating set

Question

Let $S$ be a numerical semigroup (https://en.m.wikipedia.org/wiki/Numerical_semigroup). Let $A$ be the minimal generating set for $S$. As standard, let $e(S)$, $m(S)$ and $F(S)$ stand respectively for $|A|$, $\min(A)$ and $\max (\mathbb N \setminus S)$.

My question is: If $F(S)>m(S)+e(S)$, then is it true that $F(S)>\max(A)$ ?