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 $$P_2\,$$, $$P_3\,$$, $$P_4, \dots$$ 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 $$k + m\,$$?

(第十一届AIME1993 第12题)