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Claim: $\tau \in (0,\pi/4)$.  Pf: $k=\lroof $k=\leftroof  2\pi / \tau \rroof \rrightroof  > 8$ is given. We seek an upper bound on the left hand side. \begin{align*}  t\cdot|uv| &\geq |uw| + t\cdot|wv| \\