Breaking pencils in two

You have 2016 sticks of the same length in a box. You pick a stick at random, break it into two equal halves, and put them back in for a total of 2017 sticks. You repeat this process of random picking and breaking indefinitely.

What is the maximum value of \(x\) such that, at any time during this process, you are guaranteed to have at least \(x\) sticks of the same length?

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