A geometry problem involving the Lambert $W$ function

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For what $x$ value can $W(x)$, $W(2x)$ and $W(3x)$ be the sides of a right triangle where $W$ denotes the Lambert $W$ function?

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For $x>0$ you have $W(3x)> W(2x)> W(x)$, then consider $W(3x)$ the hypotenuse, then you solve: $$W(x)^2+W(2x)^2=W(3x)^2\;\Rightarrow x\approx 1.36627870097301...$$

Then in reality there would also be complex values ​​if you change the hypotenuse to one of the other 3 values