Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ ($x, y \in \mathbb{R}$, $i = \sqrt{-1}$) of the equation $sz + t\bar{z} + r = 0$, where $\bar{z} = x - iy$. Then, which of the following statement(s) is (are) TRUE?

$\text{Given: } s, t, r \text{ non-zero complex; } sz + t\bar{z} + r = 0$ $z = x + iy,\quad x, y \in \mathbb{R},\quad \bar{z} = x - iy$ $\text{To find: which of the four statements are TRUE}$ $sz + t\bar{z} + r = 0$ $\bar{s}\,\bar{z} + \bar{t}\,z + \bar{r} = 0$ $sz + t\bar{z} = -r$ $\bar{t}z + \bar{s}\bar{z} = -\bar{r}$ $\Delta = s\bar{s} - t\bar{t} = |s|^2 - |t|^2$ $\text{If } |s| \neq |t|:\ \Delta \neq 0 \Rightarrow \text{unique solution}$ $\Rightarrow L \text{ has exactly one element — statement (1) TRUE}$ $\text{If } |s| = |t|:\ \Delta = 0 \Rightarrow \text{no unique solution}$ $\text{consistent} \Rightarrow \text{line};\quad \text{inconsistent} \Rightarrow \text{empty}$ $s = t = 1,\ r = i:\ 2x = -i\ (\text{impossible}) \Rightarrow L = \varnothing$ $|s| = |t| \not\Rightarrow \text{infinitely many elements}$ $\Rightarrow \text{statement (2) is FALSE}$ $L \text{ is } \varnothing,\ \text{a single point, or a line in } \mathbb{C}$ $\text{more than one element} \Rightarrow \text{line} \Rightarrow \text{infinitely many — (4) TRUE}$ $\{z : |z - 1 + i| = 5\} = \text{circle, radius } 5,\ \text{centre } 1 - i$ $L \cap \text{circle has at most } 2 \text{ points — (3) TRUE}$ $\text{TRUE: } (1),\ (3),\ (4)$