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Algebra Level 4

Let \(a\), \(b\), \(c\), \(d\) be real numbers and all zeroes \(\beta_1, \beta_2, \beta_3,\) and \(\beta_4\) of the polynomial \(P(x)=x^4+ax^3+bx^2+cx+d\) are real. Find the smallest value the product \((\beta_1^2+1)(\beta_2^2+1)(\beta_3^2+1)(\beta_4^2+1)\) can take.

This is a modified AIME problem.

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