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If alpha is an integer satisfying |alph...

If `alpha` is an integer satisfying `|alpha|lt=4-|[x]|,` where `x` is a real number for which `2xtan^(-1)x` is greater than or equal to `ln(1+x^2),` then the number of maximum possible values of `alpha` (where [.] represents the greatest integer function) is_____

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