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For $1\leq i\leq 215$ let $a_i=\frac{1}{2^i}$ and $a_{216}=\frac{1}{2^{215}}$. Let $x_1,x_2,\ldots,x_{216}$ be positive real numbers such that \sum\limits_{i=1}^{216} x_i=1 \text{  and  } \sum\limits_{1\leq i$x_2=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.