Let \( \left[ \begin{matrix} 2 & 1 \\ 1 & 0 \end{matrix} \right]^n=\left( a_{ij}(n) \right) \)

If \( \left( \displaystyle \lim_{n \to \infty} \dfrac{a_{12}(n)}{a_{22}(n)} \right)^2=\sqrt{A}+\sqrt{B} \quad \quad \left( A,B \in \mathbb{N}\right) \)

then find the value of \( A+B \).

**Notation**: \(a_{ij}(n)\) denotes the element in the \(i^{\text{th}}\) row and \(j^{\text{th}}\) column of matrix \(A\).

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