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How to rewrite multivariate polynomial as polynomial on one variable?

Suppose i have declared many varibles and a polynomial using them

x, y, z = var("x y z")
poly =  x^3*y*z + x^2*y^2 + 3*x^3 + 2*x^2 + x*y + x*z + 1

How can i simplify the expression is such a way that it is written as a polynomial over x? I mean something like

(...)*x^3 + (...)*x^2 + (...)*x +...