  An isoscalar $pn$ pair is expected to emerge in nuclei that have similar
proton and neutron numbers and it may be a candidate for a deuteron
``cluster.'' There is, however, no experimental evidence for it. The purpose of
this paper is to construct a new reaction model for the ($p,pd$) reaction
including the deuteron breakup in the elementary process and the deuteron
reformation by the final-state interactions (FSIs). How these processes
contribute to the observables of the reaction is investigated. The distorted
wave impulse approximation is extended in twofold. The elementary processes of
the ($p,pd$), i.e., the $p$-$d$ elastic scattering and $d(p,p)pn$ reaction, are
described with an impulse picture employing a nucleon-nucleon effective
interaction. The three-body scattering waves in the final state of the ($p,pd$)
reaction are calculated with the continuum-discretized coupled-channels method.
The triple differential cross section (TDX) of the ($p,pd$) reaction is
calculated with the new model. The elementary processes are described
reasonably well with the present model. As for the ($p,pd$) reaction, the
deuteron reformation can either increase or decrease the TDX height depending
on the interference between the elastic and breakup channel of deuteron, while
the \textit{back-coupling} effect always decreases it. It is shown that the
deuteron reformation significantly changes the TDX of the ($p,pd$) reaction
through the interference. It is important to include this process to
quantitatively discuss the ($p,pd$) cross sections in view of the deuteron
formation in nuclei. For more quantitative discussion regarding the
experimental data, further improvement will be necessary.
