  In this work we consider a cosmological scenario in which the Universe
contracts initially having a bouncing-like behavior, and accordingly after it
bounces off, it decelerates following a matter dominated like evolution and at
very large positive times it undergoes through an accelerating stage. Our aim
is to study such evolution in the context of $F(R)$ gravity theory, and
confront quantitatively the model with the recent observations. Using several
reconstruction techniques, we analytically obtain the form of $F(R)$ gravity in
two extreme stages of the universe, particularly near the bounce and at the
late time era respectively. With such analytic results and in addition by
employing appropriate boundary conditions, we numerically solve the $F(R)$
gravitational equation to determine the form of the $F(R)$ for a wide range of
values of the cosmic time. The numerically solved $F(R)$ gravity realizes an
unification of certain cosmological epochs of the universe, in particular, from
a non-singular bounce to a matter dominated epoch and from the matter dominated
to a late time dark energy epoch. Correspondingly, the Hubble parameter and the
effective equation of state parameter of the Universe are found and several
qualitative features of the model are discussed. The Hubble radius goes to zero
asymptotically in both sides of the bounce, which leads to the generation of
the primordial curvature perturbation modes near the bouncing point.
Correspondingly, we calculate the scalar and tensor perturbations power spectra
near the bouncing point, and accordingly we determine the observable quantities
like the spectral index of the scalar curvature perturbations, the
tensor-to-scalar ratio, and as a result, we directly confront the present model
with the latest Planck observations. Furthermore the $F(R)$ gravity dark energy
epoch is confronted with the Sne-Ia+BAO+H(z)+CMB data.
