  For any proper action of a non-elementary group $G$ on a proper geodesic
metric space, we show that if $G$ contains a contracting element, then there
exists a sequence of proper quotient groups whose growth rate tends to the
growth rate of $G$. Similar statements are obtained for a product of proper
actions with contracting elements.
  The tools involved in this paper include the extension lemma for the
construction of large tree, the theory of rotating families developed by F.
Dahmani, V. Guirardel and D. Osin, and the construction of a quasi-tree of
metric spaces introduced by M. Bestvina, K. Bromberg and K. Fujiwara. Several
applications are given to CAT(0) groups and mapping class groups.
