Too much pi

Calculus Level 4

\[\large \lim_{n\to \infty }\left[ \dfrac{n}{2} \left( 2\sqrt[n]{\dfrac{\pi }{2} } -\sqrt[n]{\dfrac{\pi }{3} } -\sqrt[n]{\dfrac{\pi }{4} } \right) \right] =\dfrac{\ln A }{B}\]

If the equation above holds true for positive integers \(A\) and \(B\), such that \(A\) is minimized, find \(A+B\).

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