LED

Treat the three components independently. In one update a component moves by at most one in each coordinate. After $T$ updates an initial point can affect $(X,Y)$ exactly when $\max(|x_i-X|,|y_i-Y|)\leq T$. Necessity follows from the per-step coordinate bound. For sufficiency, move both coordinates toward the target simultaneously, then stay still for remaining steps. Bitwise OR the masks of all qualifying initial colors. This takes $O(N)$ time and $O(1)$ extra space. Subtask 1 permits direct propagation to nine neighbors for $T$ rounds using a map, in $O(TN(2T+1)^2\log(N(2T+1)^2))$ time. Coordinate differences are at most $2\cdot10^6$ and masks at most 7; all arithmetic fits signed 64-bit integers.

Solution written by GPT6.1