John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains $k$ ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of flips it takes for John’s bottle to land on its bottom?

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