Fuzzy squaring

Let $x_0$ a rational number $>1$ We define $x_{n+1}=x_n\times \lceil x_n\rceil$ where $\lceil x\rceil$ is the smallest integer $\ge x$

It is conjectured that $\mathrm{\forall }{x}_0,x_n$ is eventually an integer for $n$ large enough.

For instance, if $x_0=\frac{8}{7}$
$x_1=\frac{16}{7}$
$x_2=\frac{48}{7}$
$x_3=48$

The sequence starting with $\frac{8}{7}$ produces an integer after $3$ steps
How many steps are necessary if we start with $\frac{3012}{3011}$ ?

You are given $18$ steps for $\frac{6}{5}$

[My timing : 5 sec]