Dicey Development

Monica has 3 die, each face of each dice is labeled with a distinct integer in the interval 1 to 6. And the sum of the numbers in 2 opposite faces is always 7.

She arranges her dice in a row such the faces that touch each other has the same numerical value. In the end, she obtains a 3-digit integer by reading the top faces of these dices.

For example, she can obtain the number 436 (as shown in the picture above) by facing the two "2" against each other and two "5" against each other. But it is impossible for her to obtain the number 635.

How many distinct 3-digit integers can she obtain by rearranging the dices in this way?

Problem source: OBMEP 2014

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