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Let \(a_0 = 1, b_0 = 2,\) and for \(n\geq 0\) \[a_{n+1} = \frac{a_n + b_n}{2},\quad b_{n+1} = \sqrt{a_{n+1}b_n}.\] If \( \displaystyle \lim_{n\to\infty } a_n b_n = \dfrac c{\pi^ d} \) for positive integers \(c\) and \(d\), find \(c+d\).

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