  In this paper we consider, in the general context of CAT(0) spaces, an
iterative schema which alternates between Halpern and Krasnoselskii-Mann style
iterations. We prove, under suitable conditions, the strong convergence of this
algorithm, benefiting from ideas from the proof mining program. We give
quantitative information in the form of effective rates of asymptotic
regularity and of metastability (in the sense of Tao). Motivated by these
results we are also able to obtain strongly convergent versions of the
forward-backward and the Douglas-Rachford algorithms. Our results generalize
recent work by Bo\c{t}, Csetnek and Meier, and Cheval and Leu\c{s}tean.
