An Euler Line Area Ratio

Let $H,P$ be two points in the plane of $\triangle ABC$, distinct from $A,B,C$, and write $\vec a=\overrightarrow{PA}$, $\vec b=\overrightarrow{PB}$, $\vec c=\overrightarrow{PC}$, $\vec h=\overrightarrow{PH}$. Suppose

and $|\overrightarrow{AH}|=1$, $|\overrightarrow{BH}|=\sqrt 2$, $|\overrightarrow{BC}|=\sqrt 3$. Let $O$ be the circumcenter of $\triangle ABC$. Find the ratio of the areas of $\triangle AOB$, $\triangle BOC$, $\triangle AOC$.