Please reason step by step, and put your final answer within \boxed{}.
Given a sequence of positive terms $\{a_n\}$ that satisfies $n-a_n=\frac{a_n}{a_{n-1}}$ for $n\geqslant 2, n\in \mathbb{N}^+$, and $a_1=\frac{1}{2}$. Let $b_n=\frac{2^{n-1}}{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.
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