Let \(S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\)\(\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}\), and \(T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}\). Then which of the following statements is (are) TRUE?

(1)
\(\begin{aligned}& (-1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}+\sqrt{2} \mathrm{n}, \mathrm{m}, \text{ where, } \mathrm{n} \in \mathbb{Z} \\ & (1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}_1+\sqrt{2} \mathrm{n}_1, \text{ where, } \mathrm{~m}_1, \mathrm{n}_1 \in \mathbb{Z} \\& \Rightarrow \mathbb{Z} \cup \mathrm{T}_1 \cup \mathrm{T}_2 \subseteq \mathrm{S}\end{aligned}\)
but \(b \sqrt{2} \in S\) for negative \(b \in \mathbb{Z}\).
So \(\mathbb{Z} \cup \mathrm{T}_1 \cup \mathrm{T}_2 \subset \mathrm{S}\)
(2)
\(\begin{aligned} & (\sqrt{2}-1)^{\mathrm{n}}=\frac{1}{(\sqrt{2}+1)^{\mathrm{n}}} < \frac{1}{2024} \\ & \Rightarrow 2024 < (\sqrt{2}+1)^{\mathrm{n}}, \text{ where, } \exists \mathrm{n} \in \mathbb{N} \\ & \Rightarrow \mathrm{T}_1 \cap\left(0, \frac{1}{2024}\right) \neq \phi\end{aligned}\)
(3)
\(\begin{aligned} & (1+\sqrt{2})^{\mathrm{n}}>2024, \text{ where, } \exists \mathrm{n} \in \mathbb{N} \\ & \Rightarrow \mathrm{T}_2 \cap(2024, \infty) \neq \phi\end{aligned}\)
(4)
\(\sin (\pi(a+b \sqrt{2})=0) \Rightarrow b=0, \text{ where, } a \in \mathbb{Z}\)
\(\Rightarrow\) Options (1), (3), (4) are Correct.