\(N\) is a 3-digit number that is a perfect square. When the first digit is increased by 1, the second digit is increased by 2, the third digit is increased by 3, the result is still a perfect square. Determine \(N\).

**Details and assumptions**:

Since digits do not exceed 9, the challenge assumes that the units digit of \(N\) to be at most 6, the tens digit of \(N\) to be at most 7 and the hundreds digit of \(N\) to be at most 8.

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