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Consider a $\Delta ABC$ with $\angle B=75^{\circ}$ . Let $H$ and $O$ be its $\text{orthocenter and circumcenter respectively}$ . $AD\bot BC$ with $D$ on $BC$ . Also $AH\times HD=6(\sqrt{3}-1)$ and circumradius of the $\Delta ABC$ is $2\sqrt{3}~units$ .

$\large{\angle HBO=\arcsin \left(\frac{P}{Q}(\sqrt{R}-\sqrt{S}) \right)}$

where $P,Q,R,S$ are integers and $P,Q$ are co prime and $R,S$ are square free.

Find $P+Q+R+S$

Original problem