How high does it go?

$\displaystyle\sum_{cyc}|\sin a \times \cos (b+c)| = \displaystyle\sum_{cyc}|\sin a \times \cos (π-a)|$ $\Rightarrow \displaystyle\sum_{cyc}|\sin a \times \cos (π-a)|=\displaystyle\sum_{cyc}|-\sin a \times \cos a|=\displaystyle\sum_{cyc}\bigg|\frac{\sin {2a}}{2}\bigg|$ Let $f(x) = \bigg|\frac{\sin {2x}}{2}\bigg|$ if $x \epsilon (0,\frac{π}{2})$ then $f(x)$ is concave. Therefore by Jensen's inequality $f(\frac{\sum a}{3}) \geq \frac{\sum f(a)}{3}$ $\Rightarrow 3 f(\frac{π}{3}) \geq \displaystyle\sum_{cyc}|\sin a \times \cos (b+c)|$ $\Rightarrow \displaystyle\sum_{cyc}|\sin a \times \cos (b+c)| \leq 3\times \frac{\sqrt3}{2 \times 2}$ Hence, $\Rightarrow \displaystyle\sum_{cyc}|\sin a \times \cos (b+c)| \leq \frac{3\sqrt3}{4}$