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Calculus Level 4

\[\lim_{n \rightarrow \infty} \left( \frac{N^{2}+N+1}{3} \right) \left( \frac{N^{6}+N^{3}+1}{3} \right) \left( \frac{N^{18}+N^{9}+1}{3} \right) \ldots \left( \frac{N^{2 \cdot 3^{n}}+N^{3^{n}}+1}{3} \right)\]

For integer \(n\), let \(N = 1+\frac{1}{3^{n}}\). Find the value of the limit above to nearest integer.

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