Play The Coin One Is Dealt

Three men in a room all covet a certain object. They agree to flip a coin for it in such a way that they all have equal chances of getting it.

Unfortunately, the only coin they can find is a biased one, rigged to land heads \(\dfrac35\) of the time and tails \(\dfrac25\) of the time. The men all know the exact odds for the biased coin.

Using the biased coin, what is the minimum number of flips in the best case needed to determine in a fair way which of the three gets the object?

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