After an extremely intense chess game, the triumphant White King decides to expand his kingdom from an \(8 \times 8\) board to an infinite grid, with all squares coloured white in his honour.

Admiring his pristine terra but desiring more colour and flair to his new land, he goes for a walk, initially facing north, following these instructions:

At a white square, turn \(90^{\circ}\) right, colour the square to red, move forward one square.

At a red square, turn \(90^{\circ}\) left, colour the square to white, move forward one square.

After \(n\) moves, the king begins moving in a fixed, orderly pattern in the southeast direction. Find \(n\), rounding your answer to the nearest thousand.


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