  We propose to consider the N=4,d=1 supermultiplet with $% (4,4,0) component
content as a ``root'' one. We elaborate a new reduction scheme from the
``root'' multiplet to supermultiplets with a smaller number of physical bosons.
Starting from the most general sigma-model type action for the ``root''
multiplet, we explicitly demonstrate that the actions for the rest of linear
and nonlinear N=4 supermultiplets can be easily obtained by reduction.
  Within the proposed reduction scheme there is a natural possibility to
introduce Fayet-Iliopoulos terms. In the reduced systems, such terms give rise
to potential terms, and in some cases also to terms describing the interaction
with a magnetic field.
  We demonstrate that known N=4 superconformal actions, together with their
possible interactions, appear as results of the reduction from a free action
for the ``root'' supermultiplet. As a byproduct, we also construct an N=4
supersymmetric action for the linear (3,4,1) supermultiplet, containing both an
interaction with a Dirac monopole and a harmonic oscillator-type potential,
generalized for arbitrary conformally flat metrics.
