An infinite line of stepping stones stretches out into an infinitely large lake.

A frog starts on the second stone.

Every second he takes a jump to a neighboring stone. He has a probability, \(p\), of jumping one stone further away from the shore and a probability, \(1-p\), of jumping one stone closer to the shore.

If the expected value for the number of jumps he will take before reaching the first stone is exactly 100, then \(p\) can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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