Crazy Coin Flipping

There are two teams, The Eagles and The Hawks, consisting of \(6\) people each.

Every person in The Eagles has a fair coin. But, every person in The Hawks has a biased coin which comes up heads with a probability of \(3/4\).

Everybody tosses their coin exactly once. The probability that The Hawks get more heads than The Eagles is \(P\).

Find \[\lfloor 1000P \rfloor\]

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