If \( x^{2}+a x+b=0 \) has two distinct negative integral roots and \( \left(\log _{1 / 2}\left(...

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If \( x^{2}+a x+b=0 \) has two distinct negative integral roots and \( \left(\log _{1 / 2}\left(\frac{2}{\pi} \cot ^{-1} x+1\right)\right)^{2}+a \log _{1 / 2}\left(1-\frac{\tan ^{-1} x}{\pi}\right)=a-b \) has no real solution. Then find the minimum value of \( a \).
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