Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.