What was Lame's proof?

In 1847, Lame gave a false proof of Fermat’s Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity.

The best description I’ve found is in the book Fermat’s Last Theorem A Genetic Introduction to Algebraic Number Theory. For the equation $x^n + y^n = z^n$, it says

That is, he planned to show that if $x$ and $y$ are such that the factors $x+y, x+ry, \dots, x+r^{n-1}y$ are relatively prime then $x^n + y^n = z^n$ implies that each of the factors $x+y, x+ry, \dots$ must itself be an $n$th power and to derive from this an impossible infinite descent. If $x+y, x+ry, \dots$ are not relatively prime he planned to show that there is a factor $m$ common to all of them so that $(x+y)/m, (x+ry)/m, \dots, (x+r^{n-1}y)/m$ are relatively prime and to apply a similar argument in this case as well.

Nowhere else can I find any more detail as to how the rest of the argument goes. I don’t see how to use infinite descent here, can anyone fill in the details?

I would like to also mention Conrad’s great notes on Kummer’s proof for regular primes. While this does indeed give a proof, it seems quite sophisticated and I am hoping there is an easier method when one assumes (falsely, in general) that $\mathbb{Z}[r]$ is a UFD.

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