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)