1993 AIME Problem 12

The vertices of \triangle A B C are A=(0,0), B=(0,420), and C=(560,0). The six faces of a die are labeled with two A's, two B's, and two C's. Point P_{1}=(k, m) is chosen in the interior of \triangle A B C, and points P_{2}, P_{3}, P_{4}, \ldots are generated by rolling the die repeatedly and applying the rule: If the die shows label L, where L \in\{A, B, C\}, and P_{n} is the most recently obtained point, then P_{n+1} is the midpoint of \overline{P_{n} L}. Given that P_{7}=(14,92), what is k+m?