Školjka

%V0 For any positive integer $x$ define $g(x)$ as greatest odd divisor of $x,$ and $$f(x) =\begin{cases}\frac{x}{2}+\frac{x}{g(x)}&\text{if\ \(x\) is even},\\ 2^{\frac{x+1}{2}}&\text{if\ \(x\) is odd}.\end{cases}$$ Construct the sequence $x_1 = 1, x_{n + 1} = f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $n$ such that $x_n = 1992,$ and determine whether $n$ is unique.