Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $$P_{i}$$ is defined recursively as follows: $$P_{0}$$ is a regular tetrahedron whose volume is 1. To obtain $$P_{i + 1}$$, replace the midpoint triangle of every face of $$P_{i}$$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $$P_{3}$$ is $$\frac {m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m + n$$.

(第十九届AIME2 2001 第12题)