Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts with $2012$. For example, $586$ is a leader since $586^3 = 201230056$. How many leaders are there in the set $\{1, 2, 3, \ldots, 2012\}$?

Let's think step by step and output the final answer within \boxed{}.
user