1993 AIME Problem 12

The vertices of \triangle ABC 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 ABC, 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\,?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.