Let $R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin\theta = y \sin\!\left(\theta + \tfrac{2\pi}{3}\right) = z \sin\!\left(\theta + \tfrac{4\pi}{3}\right)$, with $\theta \in (0, 2\pi)$. For a square matrix $M$, let $\text{Trace}(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) $\text{Trace}(R) = 0$

(II) If $\text{Trace}(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.

(1) Both (I) and (II) are true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Only (I) is true