The Grid: A digital frontier

In the early days of the construction of the grid, Flynn decided to build an \(n\times n\) grid such that he want his robots to travel from one end of the grid to the other opposite end of the grid. And his robots are programmed to travel either 1 unit up or 1 unit to the right.

However, the robots he created are still buggy because they can only travel either 2 units up or 2 units to the right. After a couple of simulation, he discovered that the total number of path one robot can make from a bottom left end of the grid to the top right end of the grid is in between 500 and 1000. What is the value of \(n\)?

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