  In this paper, we study (1,2) and (2,1) random walks in varying environments
on the lattice of positive half line. We assume that the transition
probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$
For (1,2) random walk, we get some elaborate asymptotic behaviours of various
escape probabilities and hitting probabilities of the walk. Such observations
and some delicate analysis of continued fractions and the product of
nonnegative matrices enable us to give criteria for finiteness of the number of
cutpoints of both (1,2) and (2,1) random walks, which generalize E. Cs\'aki, A.
F\"oldes and P. R\'ev\'esz [J. Theor. Probab. 23: 624-638 (2010)] and H.-M.
Wang [Markov Processes Relat. Fields 25: 125-148 (2019)]. For near-recurrent
random walks, whenever there are infinitely many cutpoints, we also study the
asymptotics of the number of cutpoints in $[0,n].$
