  We report on the first results of a full three-body calculation of the
$\bar{K}NN$-$\pi YN$ amplitude for the $K^-d\rightarrow\pi\Sigma n$ reaction,
and examine how the $\Lambda(1405)$ resonance manifests itself in the neutron
energy distributions of $K^-d\rightarrow\pi\Sigma n$ reactions. The amplitudes
are computed using the $\bar{K}NN$-$\pi YN$ coupled-channels
Alt-Grassberger-Sandhas (AGS) equations. Two types of models are considered for
the two-body meson-baryon interactions: an energy-independent interaction and
an energy-dependent one, both derived from the leading order chiral SU(3)
Lagrangian. These two models have different off-shell properties that cause
correspondingly different behaviors in the three-body system. As a remarkable
result of this investigation, it is found that the neutron energy spectrum,
reflecting the $\Lambda(1405)$ mass distribution and width, depends quite
sensitively on the (energy-dependent or energy-independent) model used. Hence
accurate measurements of the $\pi\Sigma$ mass distribution have the potential
to discriminate between possible mechanisms at work in the formation of the
$\Lambda(1405)$.
