  It has been recently demonstrated that the thermal partition function of any
large $N$ Chern-Simons gauge theories on $S^2$, coupled to fundamental matter,
reduces to a capped unitary matrix model. The matrix models corresponding to
several specific matter Chern-Simons theories at temperature $T$ were
determined in arXiv:1301.6169. The large $N$ saddle point equations for these
theories were determined in the same paper, and were solved in the low
temperature phase. In this paper we find exact solutions for these saddle point
equations in three other phases of these theories and thereby explicitly
determine the free energy of the corresponding theories at all values of
$T^2/N$. As anticipated on general grounds in arXiv:1301.6169, our results are
in perfect agreement with conjectured level rank type bosonization dualities
between pairs of such theories.
