Pratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, while Payel multiplies his number by $2$ and then adds $2$ to it. In other words, on the first day, their numbers will be $(2n - 2)$ and $(2m + 2)$ respectively. Find the minimum integer $x$ with proof such that if $n - m \geq x$, then Pratyya's number will be larger than Payel's number every day.

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