Convergence of a Custom Die

Imagine rolling a standard 6-sided die six times, noting each outcome. You then inscribe these results onto the faces of a new, blank die. For instance, if you rolled: 3, 5, 3, 6, 1, 2 The second die would lack a 4, instead featuring two 3s. You then roll this newly created die six times, using those results to label the faces of another fresh die. This iterative process continues, crafting a new die from its predecessor's outcomes. Eventually, you'll craft a die where all six faces display the same number. What method would you use to calculate the average number of iterations needed to achieve this uniformity?