Let he who has no sin cast the first stone

Calvin and Lino are playing a game with 4 piles of stones. At the start of the game, Calvin rolls four 8-sided dice, each with the numbers 0 through 7 on them, to help in determining the sizes of the piles of stones. Pile \(i\) will have size \(10^i + d_i,\) where \(d_i\) is the number rolled on the \(i\)th die.

Players take turns removing stones from the piles, and Lino gets to make the first move. On a player's turn, that player is allowed to remove a number of stones that is a power of 2 from one of the piles. The last player who makes a move is the winner.

Assuming that Calvin and Lino both play optimally, the probability that Lino is able to remove 4 stones from a pile on his first move and still win the game can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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