Question 214, Find all $x \in[1,2)$ such that for any positive integer $n,\left[2^{\mathrm{n}} x\right]$ modulo 4 is 1 or 2.

保留源文本的换行和格式，翻译结果如下：

Question 214, Find all $x \in[1,2)$ such that for any positive integer $n,\left[2^{\mathrm{n}} x\right]$ modulo 4 is 1 or 2.
Question 214, Solution: Let $x=\overline{\left(1 . a_{1} a_{2} \ldots\right)_{2}}$, then for any positive integer $n$, we have $\left[2^{n} x\right]=\overline{\left(1 a_{1} a_{2} \ldots a_{n}\right)_{2}} \equiv 2 a_{n-1}+a_{n}(\bmod 4)$. In particular, $a_{0}=1$.
Thus, we know $a_{1}=0, a_{2}=1, a_{3}=0, a_{4}=1, \cdots$. Therefore, $x=\overline{(1.0101 \ldots)_{2}}=\frac{4}{3}$.