Weakly finitely presented infinite periodic groups

A group $G$ given by a presentation $G=<\mathcal{A}\mathrm{\parallel }\mathcal{R}>$ is called weakly finitely presented if every finitely generated subgroup of $G$, generated by (images of) some words in ${\mathcal{A}}^{\pm 1}$, is naturally isomorphic to the subgroup of a group $G_0=<{\mathcal{A}}_0\mathrm{\parallel }{\mathcal{R}}_0>$, where ${\mathcal{A}}_0\subseteq \mathcal{A}$, ${\mathcal{R}}_0\subseteq \mathcal{R}$ are finite, generated by (images of) the same words. In the article, weakly finitely presented periodic groups which are not locally finite are constructed.