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There is a problem ...
Simplest is always "this is wakening #1". 50% chance of escape, and soonest possible if it happens. Has the psychological disadvantage that if the first coin is tails, you're stuck forever with no future chance of escape. You have no memory of any of them, so that is irrelevant - all that matters is the probability of escape - but it feels bad to us as outside observers.
You can stretch it out by randomizing your answer with a declining chance of higher numbers. Say, flip YOUR coin until you get tails, then guess the number of flips you've mad... (read more)