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Algebra Level 4

$x^4+3x^3+4x^2+3x+1=0$

Let $a,b,c,$ and $d$ be the not necessarily distinct roots of the equation above. Find the value of the expression below:

$a^{1024}+b^{1024}+c^{1024}+d^{1024}+\frac{1}{a^{1024}}+\frac{1}{b^{1024}}+\frac{1}{c^{1024}}+\frac{1}{d^{1024}}.$