SMO 2006 Open Round 1 Question 11

In triangle $ABC$, $AB=AC=1$, and $D,E$ are the midpoints of $AB,AC$, respectively. Let $P$ be a point on $DE$. The extensions of $BP$ and $CP$ meet the sides $AC$ and $AB$ at $G$ and $F$, respectively. Find $\dfrac1{BF}+\dfrac1{CG}$.