Construction of a valued field whose valuation ring is definable but neither $\mathrm{\exists }\mathrm{\forall }\mathrm{\exists }$ nor $\mathrm{\forall }\mathrm{\exists }\mathrm{\forall }$-definable in the language of rings

We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\mathrm{\exists }\mathrm{\forall }\mathrm{\exists }$ or $\mathrm{\forall }\mathrm{\exists }\mathrm{\forall }$-formula in $L_{\text{ring}}$ to define it.