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)