\[ \large f_n(x)=x^2 + ( x + 1) ^2 +(x+2)^2+\ldots+(x+n)^2 \]

The function \(f_n(x) \) is defined above for \(x\) ranges over all real values with natural number \(n\).

If the minimum value of \(f_n(x)\) is \((2^{10}-12)\) then find the value of \(n\).

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