Wu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, Wu receives a coin, and for each tails he flips, Wu loses a coin. He continues this process each year until he has $0$ coins, at which point he stops. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a$ and $b$ are positive integers with $a$ being odd. Find $a+b$.

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