2001 AIME II Problem 12

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