  We observed an X-ray afterglow of GRB 060904A with the Swift and Suzaku
satellites. We found rapid spectral softening during both the prompt tail phase
and the decline phase of an X-ray flare in the BAT and XRT data. The observed
spectra were fit by power-law photon indices which rapidly changed from $\Gamma
= 1.51^{+0.04}_{-0.03}$ to $\Gamma = 5.30^{+0.69}_{-0.59}$ within a few hundred
seconds in the prompt tail. This is one of the steepest X-ray spectra ever
observed, making it quite difficult to explain by simple electron acceleration
and synchrotron radiation. Then, we applied an alternative spectral fitting
using a broken power-law with exponential cutoff (BPEC) model. It is valid to
consider the situation that the cutoff energy is equivalent to the synchrotron
frequency of the maximum energy electrons in their energy distribution. Since
the spectral cutoff appears in the soft X-ray band, we conclude the electron
acceleration has been inefficient in the internal shocks of GRB 060904A. These
cutoff spectra suddenly disappeared at the transition time from the prompt tail
phase to the shallow decay one. After that, typical afterglow spectra with the
photon indices of 2.0 are continuously and preciously monitored by both XRT and
Suzaku/XIS up to 1 day since the burst trigger time. We could successfully
trace the temporal history of two characteristic break energies (peak energy
and cutoff energy) and they show the time dependence of $\propto t^{-3} \sim
t^{-4}$ while the following afterglow spectra are quite stable. This fact
indicates that the emitting material of prompt tail is due to completely
different dynamics from the shallow decay component. Therefore we conclude the
emission sites of two distinct phenomena obviously differ from each other.
