The maximum value of the function defined by \( f(x)=\min \left(e^{x}, 2+e^{2}-x, 8\right) \) is...

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The maximum value of the function defined by \( f(x)=\min \left(e^{x}, 2+e^{2}-x, 8\right) \) is a, then integral value of \( x \) satisfying the inequality \( \frac{x(x-[a])}{x^{2}-[a] x+12}<0 \), is :
[Note : \( [k] \) denotes greatest integer function less than or equal to \( k \).]
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