Question 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 $\overline{P_nL}.$ Given that $P_7=(14,92),$ what is $k+m?$