All Values Represented

How many distinct $6-$ sided dice with values in the set $[6] = {1,2,\dots, 6}$ are there such that the following property holds: When this die is rolled twice, there is positive probability that the sum of the upfaces is $k$ for each $2 \leq k \leq 12$ . Two dice are considered indistinguishable if the number of faces that show value $i$ for $1 \leq i \leq 6$ is the same for all $i$ between the two dice, regardless of orientation. Assume that each of the $6$ sides appear with equal probability.

Answer format: enter an integer.