A plane passing through the vertex $A$ and the center of its inscribed sphere of a tetrahedron $ABCD$ intersects its edge $BC$ and $CD$ at point $E$ and $F$, as shown. If $AEF$ divides this tetrahedron into two equal volume parts: $A-BDEF$ and $A-CEF$, what is the relationship between these two parts' surface areas $S_1$ and $S_2$ where $S_1 = S_{A-BDEF}$ and $S_1=S_{A-CEF}$? $(A) S_1 < S_2\quad(B) S_1 > S_2\quad (C) S_1 = S_2 \quad(D) $ cannot determine