Let \(\Gamma\) be a circle with radius \(r\). Let \(A\) be any point on \(\Gamma\) and let \(t\) be the tangent to \(\Gamma\) at point \(A\). Let \(B\) and \(C\) be points of \(t\) on opposite sides of \(A\) such that \(AB = 12r\) and \(AC = 24r\). Let \(P\) be any point of \(\Gamma\) different from \(A\). Find the value of:

\[\large{\dfrac{1}{\cot \angle APB +\cot \angle APC}}\]

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