\[ \large \left (\prod_{k=6}^{1729} P_k N_k + \prod_{k=6}^{1729} Z_k \right )\bmod {11}\]

Let \( f_k(x) = 1729x^k + 8x^{k-1} - x^3 + kx - 1 \) for integer \(6\leq k\leq 1729\).

We further define these

- \(P_k\) as the number of maximum possible positive roots of \(f_k(x) \).

- \(N_k\) as the number of maximum possible negative roots of \(f_k(x) \).

- \(Z_k\) as the number of maximum possible non-real roots of \(f_k(x) \).

Evaluate the modulo of the sum of the products given above.

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