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We present an asymptotic analysis of the Verblunsky coefficients for thepolynomials orthogonal on the unit circle with the varying weight $e^{-nV(\cosx)}$, assuming that the potential $V$ has four bounded derivatives on $[-1,1]$and the equilibrium measure has a one interval support. We obtain theasymptotics as a solution of the system of "string" equations.

Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight