Say now that we have 4 distinct dice each with 7 faces labelled \( 1,2,3,4,5,6,7 \), and I roll all 4 simultaneously. Again let \( d_{1} \) be the number that comes up on the first dice, \( d_{2} \) be the number that comes up on the second dice, \( d_{3} \) be the number that comes up on the third dice and \( d_{4} \) be the number that comes up on the fourth dice. In how many ways can I roll the 4 dice in this way so that 3 divides \( d_{1} + d_{2} + d_{3} + d_{4} \)?
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For the easier version, see dice sums 1.

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