Eccoci.

A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for a total of $54$ cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately $29.05361725$ cards so that we have at least one card for each rank.

Now, assume you have $10$ such decks, each with a different back design. We shuffle all $10 \times 54$ cards together and draw cards without replacement. What is the expected number of cards needed so every suit, rank and deck design have at least one card?

Give your answer rounded to eight places after the decimal point.