  The vacuum $R^2$ model is known to generate a quasi-de Sitter evolution for
inflation, using solely the slow-roll assumptions. Using standard
reconstruction techniques, we demonstrate that the $f(R)$ gravity which
actually realizes the quasi-de Sitter evolution is not simply the $R^2$ model
but a deformed $R^2$ model which contains extra terms in addition to the $R^2$
model. We analyze in detail the inflationary dynamics of the deformed $R^2$
model and we demonstrate that the predictions are quite close to the ones of
the pure $R^2$ model, regardless the values of the free parameters. Basically
the deformed $R^2$ model is also a single parameter inflationary model, exactly
like the ordinary $R^2$ model. In contrast to the early-time era, where the
deformed $R^2$ model is quite similar to the $R^2$ model, at late times, the
phenomenological picture is different. The deformed $R^2$ model describes
stronger gravity compared to the ordinary $R^2$ with an effectively smaller
effective Planck mass. We propose the addition of an early dark energy term
which does not affect at all the inflationary era, but takes over the control
of the late-time dynamics at late times. We study in some detail the predicted
dark energy era evolution, and we demonstrate that the dark energy corrected
deformed $R^2$ model can describe a viable dark energy era, compatible with the
Planck constraints on cosmological parameters. Furthermore the model is
distinct from the $\Lambda$-Cold-Dark-Matter model, but shows a qualitatively
similar behavior.
