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How to find a polynomial identity to evaluate sum of fifth powers?

I saw the following problem on a book of mine:

Let $a,b,c\in \mathbb C$ satisfies $a+b+c=3,a^2+b^2+c^2=5, a^3+b^3+c^3=7$. Compute $a^5+b^5+c^5$.

How one can do it by Sage? I found a mathematician who was able to find the identity that solves the problem.

sage: a = var('a')
sage: b = var('b')
sage: c = var('c')
sage: x = a+b+c
sage: y = a^2+b^2+c^2
sage: z = a^3+b^3+c^3
sage: expand((x^5-5*x^3*y+5*x^2*z+5*y*z)/6)
a^5 + b^5 + c^5
sage: expand((3^5-5*3^3*5+5*3^2*7+5*5*7)/6)

But can I use Sage to find the correct identity?