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One way to efficiently navigate through a grid, covering all required squares, is to use a technique called "depth-first search" (DFS).

DFS works by starting at a specific square in the grid and exploring as far as possible along each path before backtracking. We can think of this as a "tree" structure, where each square in the grid is a node in the tree and the possible moves to adjacent squares are the edges.

To implement DFS, we can use a stack to keep track of the nodes to be explored. We start by pushing the starting square onto the stack. Then, while the stack is not empty, we pop a node from the stack, mark it as visited, and push its unvisited neighbors onto the stack. We continue this process until we reach a dead end (i.e., no unvisited neighbors) and then backtrack to the previous node on the stack.

By using DFS, we can efficiently cover all required squares in the grid because it explores each path as far as possible before backtracking, thus minimizing the number of redundant visits to the same squares.