Probability of rolling at least one 6 while rerolling 1's

Probability of rolling at least one 6 while rerolling 1's

Scenario: You roll a number of 6-sided dice
Success: Roll at least one 6
Conditions: You can re-roll any 1's you get on the first roll
What are the odds of success for n dice?
Example: 7 dice
Roll: (1, 1, 1, 4, 4, 4, 5)
Re-roll: (1, 2, 3)
Final dice: (1, 2, 3, 4, 4, 4, 5)
Result: Failure
My take on this was: For any number of rolled 1's, "replace" those rolls
with rolling n+x dice (where x is the number of 1's rolled) and thus
reducing the problem to simple combinatorics, but I didn't get very far.
I suppose there is a simple "trick", so I'm looking for other angles into
this problem.
However, if it turns out not to be so simple, please try to be as verbose
and layman-friendly as you can.