For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. Once a sapling is turned into a perfectly-shaped apple tree, it will remain so. 

The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Let's think step by step and output the final answer within \boxed{}.
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