Equilibrium redraw threshold in a uniform duel

Two players each privately draw a value from $U(0,1)$. Each may, without the other knowing, discard their value and take a fresh $U(0,1)$ draw instead, which they must keep. The higher final value wins. In equilibrium both players use a rule of the form "redraw if below $k$". Find $k$ to three decimal places.

$k=\tfrac{\sqrt5-1}{2}\approx0.618$. Against an opponent using threshold $q$, compute your winning probability as a function of your own threshold $p$ by splitting on who redraws. In equilibrium the player holding exactly $k$ must be indifferent between keeping it and redrawing, which reduces to $k^2+k-1=0$.