Having completed 11 of the 12 labours, Hercules finally faces the last and most dangerous labour, the Hydra. The only attack Hercules has for the beast is to sever its head off.

However, every time the Hydra's head is cut, there is equal chance of it remaining beheaded or sprouting 5 monstrous heads.

Fortunately for the hero, the Hydra has only 1 head to start with. Assuming the probability that Hercules finally defeats this creature of darkness be \(P\). Find the value of

\[\lfloor 10^{10} P \rfloor\]

- I am terribly sorry for having almost completely duplicated this problem. It as never my intention and I did not know about the problem. Sorry.

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