Some semi-stochastic simulations with sample size 100 and N = 2, 3, and 4 tell me that the optimal strategy consists of
Some strong contenders:
You can play around online here - further runs at higher sample sizes with teams of all kinds shows that it's actually pretty easy to get above even against [uniformly] random teams, but hard to get a winrate of ....
OPTIMAL STRATEGY to escape in less than 2^n days
I proved above that it is always possible to escape in less than 2^n days with optimal play. Here I will also show 'how' to do it (strategy), and prove that it is optimal and always possible.
Generate ordered list of 'win/optimal positions' ( arrangements of n H/T coins ):
starts with HH..HH at ...
You can escape in LESS than 2n days
It was already proven that you can ALWAYS escape in 2n "cycles" (each cycle has n days) which means escape is always possible in n*2n days (but that is not optimal play).
Optimal play require that you never repeat same position (arrangement of the coins) - because that would mean Devil has won since it will ...