For convenience, let's pretend that there are two extra vertical lasers with ~V~ values ~0~ and ~1\,000\,000~, and similarly two extra horizontal lasers with ~H~ values ~0~ and ~1\,000\,000~. Then, let ~x_1~ be the ~x~-coordinate of the nearest vertical laser to the left of Ethan (in other words, the largest ~V~ value smaller than ~X_E~). Similarly, let ~x_2~ be the ~x~-coordinate of the nearest vertical laser to the right of Ethan (the smallest ~V~ value larger than ~X_E~), and let ~y_1~ and ~y_2~ be the ~y~-coordinates of the nearest horizontal lasers below and above Ethan, respectively. The values ~x_1~, ~x_2~, ~y_1~, and ~y_2~ may be computed by iterating over all of the lasers' ~H~ and ~V~ values, in ~\mathcal O(N+M)~ time.

We can then observe that the ~i~-th microchip is reachable if and only if ~x_1 < X_i < x_2~ and ~y_1 < Y_i < y_2~. This check can be performed in ~\mathcal O(1)~ time per microchip, resulting in a total time complexity of ~\mathcal O(N+M+K)~.