Ancient Magic

There is a game played on a one-dimensional axis. $N$ stones will appear on this axis. Each of these stones is described by three numbers $x_i$, $t_i$, $c_i$, meaning the stone will appear at a point $x_i$, at time $t_i$, and has a value $c_i$. There are $p$ players initially (at time $0$) all at the point $0$. If a player is at a point $x$ at time $t$, then at time $t + 1$, he/she can be either