Let $S=\{a+b \sqrt{2}: a, b \in Z\}, T_1=\left\{(-1+\sqrt{2})^n: n \in N\right\}$ and $T_2=\left\{(1+\sqrt{2})^n: n \in N\right\}$. Then which of the following statements is (are) TRUE?
Select ALL correct options:
A
$Z \cup T_1 \cup T_2 \subset S$
B
$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set
C
$T_2 \cap(2024, \infty) \neq \phi$
D
(D) For any given $a, b \in Z, \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$, where $i=\sqrt{-1}$
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