APMO 2012 - Asian Pacific Mathematics Olympiad Question 1

Let $P$ be a point inside triangle $ABC$. Let $D,E,F$ be the intersections of line $AP$ with side $BC$, line $BP$ with side $CA$, and line $CP$ with side $AB$, respectively. Prove that the area of triangle $ABC$ is 6 if each of triangles $PFA,PDB,PEC$ has area 1.

Newly drawn configuration: