Evaluate: \(\tan^{-1}(1) + \tan^{-1}(4) + \tan^{-1}(5) + \tan^{-1}\left(\frac{1}{4}\right) = \pi + \tan^{-1}\left(\frac{\alpha}{2}\right)\). Find the value of \(\alpha\).
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Always look for reciprocal pairs \((x, 1/x)\) in \(\tan^{-1}\) sums. Their sum is always \(\pi/2\). This is a favorite trick in entrance exams to test your observation skills.