\[\large \lim_{n\to\infty} \bigg ( 4^n (3-\sqrt{3+3T_n} ) \bigg ) \]

Consider the recurrence relation \(T_n = \sqrt{2+T_{n-1}} \) with \(T_0 = 0 \). If the limit above can be expressed as \( A \pi^B \) for rational numbers \(A\) and \(B\), find the value of \(B\div A\).

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