Implement Kac ring with O(1) color and kstep

You are implementing a Kac ring dynamical system.

Model

• There are N positions arranged in a circle, indexed clockwise: 0, 1, ..., N-1 .
• Each position holds exactly one ball, whose color is either white or black .
• A subset of positions is marked .
• Initially ( time t = 0 ), all balls are white .

One time step

At each step:

1. Every ball moves one position clockwise : a ball at position i moves to position (i + 1) mod N .
2. If a ball leaves a marked position (i.e., it was at a marked index before moving), it flips color (white ↔ black) as it moves.

Task

Implement a class/data structure with the following interface:

Initialization

KacRing(N, marked)

• N : number of positions.
• marked : a sorted list of distinct marked indices 0 <= y1 < ... < ym < N .

Methods

1. step()
   • Advances the system by exactly 1 time step.
2. kstep(k)
   • Advances the system by k time steps.
   • Must run in O(1) time per call (i.e., it must not loop for k steps).
3. color()
   • Returns the value:

$\frac{W - B}{N}$

 where `W` is the current number of white balls and `B` is the current number of black balls.

• Must run in O(1) time.

Notes / Constraints

• Assume 1 <= N .
• 0 <= len(marked) <= N .
• k can be very large (e.g., up to 10^18 ), so kstep cannot simulate step-by-step.
• You do not need to expose individual ball colors; only the required methods above.