  We consider $U(N)_k$ Chern-Simons theory on $S^3$ in Seifert framing and
write down the partition function as a unitary matrix model. In the large $k$
and large $N$ limit the eigenvalue density satisfies an upper bound
$\frac{1}{2\pi\lambda}$ where $\lambda=N/(k+N)$. We study the partition
function under saddle point approximation and find that the saddle point
equation admits a gapped solution for the eigenvalue density. The on-shell
partition function on this solution matches with the partition function in the
canonical framing up to a phase. However the eigenvalue density saturates the
upper cap at a critical value of $\lambda$ and ceases to exist beyond that. We
find a new phase (called cap-gap phase) in this theory for $\lambda$ beyond the
critical value and see that the on-shell free energy for the cap-gap phase is
less than that of the gapped phase. We also check the level-rank duality in the
theory and observe that the level-rank dual of the gapped phase is a
\emph{capped} phase whereas the cap-gap phase is level-rank dual to itself.
