If \( \alpha \) is an integer satisfying \( |\alpha| \leq 4-|[x]| \...

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If \( \alpha \) is an integer satisfying \( |\alpha| \leq 4-|[x]| \), where \( x \) is a real number for which \( 2 x \tan ^{-1} x \) is greater than or equal to
\( \mathrm{P} \) \( \ln \left(1+x^{2}\right) \), then the number of maximum possible values of \( \alpha \)

W (where [.] represents the greatest integer function) is
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