6 Dice Arrangement

Six standard six-sided dice are rolled. Let \(p\) be the probability that the dice can be arranged in a row such that for \(1 \leq k \leq 6\) the sum of the first \(k\) dice is not a multiple of 3. Then \(p\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

You get to choose the order that the dice is arranged. For example, if the dice thrown was \( \{ 1, 1, 1, 2, 2, 3\}\), then we can reorder them as \( ( 1, 1, 2, 1, 2, 3) \) which satisfies the conditions.

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