If you roll five six-sided dice, then put aside any one set of same numbered dice/die that show the/a most occurring number and re-roll any remaining dice/die, what is the chance that within those two rolls (or one roll if all the dice showed the same number on the first roll) all the dice will show the same number?

To clarify:

If you roll 1, 1, 1, 4 & 6 on the first roll, you will re-roll the dice that show 4 & 6 on the second roll.

If you roll 1, 2, 2, 5 & 5 on the first roll, you will re-roll either the dice that show 1 & 2 or the dice that show 1 & 5 on the second roll as 2 and 5 are both most occurring numbers.

If you roll 3, 3, 4, 4 & 4 on the first roll, you will re-roll the dice that show 3 on the second roll as 4 is more commonly shown.

If you roll 5, 5, 5, 5 & 5 on the first roll, you will not re-roll any of the dice as they already all show the same number.

If you roll 1, 2, 3, 4 & 6 on the first roll, you will re-roll all except one of the dice on the second roll as all of the numbers present are most occurring numbers (even though it would make no difference to the chance of all the dice showing the same number if you were to just re-roll all of them).