As shown in the figure, in triangle ABC, point D is taken on segment AB such that $\overline{\mathrm{AD}} : \overline{\mathrm{DB}} = 3 : 2$, and a circle $O$ centered at A passing through D intersects segment AC at point E. $\sin A : \sin C = 8 : 5$, and the ratio of the areas of triangles ADE and ABC is $9 : 35$. When the circumradius of triangle ABC is 7, what is the maximum area of triangle PBC for a point P on circle $O$? (Given: $\overline{\mathrm{AB}} < \overline{\mathrm{AC}}$) [4 points] (1) $18 + 15\sqrt{3}$ (2) $24 + 20\sqrt{3}$ (3) $30 + 25\sqrt{3}$ (4) $36 + 30\sqrt{3}$ (5) $42 + 35\sqrt{3}$
As shown in the figure, in triangle ABC, point D is taken on segment AB such that $\overline{\mathrm{AD}} : \overline{\mathrm{DB}} = 3 : 2$, and a circle $O$ centered at A passing through D intersects segment AC at point E.\\
$\sin A : \sin C = 8 : 5$, and the ratio of the areas of triangles ADE and ABC is $9 : 35$. When the circumradius of triangle ABC is 7, what is the maximum area of triangle PBC for a point P on circle $O$?\\
(Given: $\overline{\mathrm{AB}} < \overline{\mathrm{AC}}$) [4 points]\\
(1) $18 + 15\sqrt{3}$\\
(2) $24 + 20\sqrt{3}$\\
(3) $30 + 25\sqrt{3}$\\
(4) $36 + 30\sqrt{3}$\\
(5) $42 + 35\sqrt{3}$