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Manually grouping symbolic terms

Given a symbolic expression like:

sage: var("a b c x y")
(a, b, c, x, y)
sage: a*x^2 + a*y^2 + b*y^2 + c*y^2 + (2*a*y + b*y)*x

How can I manually collect terms together? I want to represent this equation in the form:

$$ Ax^2 + Bxy + Cy^2 $$

Where $A = a, B = 2a + b, C = a + b + c$. Desired output should be:

a*x^2 + (2*a + b)*x*y + (a + b + c)*y^2

And then is there any way to read off these coefficients? Thanks