Christmas Streak 10/88: Two Strange To Be Strange

I have two distinct, fair, six-sided dice that have positive integers on their sides. Neither of the dice is a "normal" die (i.e., with 1, 2, 3, 4, 5, and 6 on the sides).

Interestingly, the probability distribution of the sum of rolling these two dice is exactly the same as that of the sum of rolling 2 "normal" dice.

Let the numbers on the first die be \((a_1, a_2, a_3, a_4, a_5, a_6),\) and those on the second \((b_1, b_2, b_3, b_4, b_5, b_6), \) where \(a_1\leq a_2 \leq\cdots\leq a_6\) and \(b_1\leq b_2 \leq \cdots\leq b_6.\)

Submit your answer as the product of these two 6-digit integers: \(\overline{a_1 a_2 a_3 a_4 a_5 a_6} \times \overline{b_1 b_2 b_3 b_4 b_5 b_6 }. \)

The dice above have 1,1,2,2,4,8 and 1,2,3,3,4,4 on their sides.  Their sum can produce the same numbers (2-12) as the sum of two ordinary dice, but not with the same probability distribution.

The dice above have 1,1,2,2,4,8 and 1,2,3,3,4,4 on their sides. Their sum can produce the same numbers (2-12) as the sum of two ordinary dice, but not with the same probability distribution.

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