An Iterated Fraction Function

Let $f_{k,t}(x)=\dfrac{kx+t}x$ (for $k,t,x\in\mathbb R$, $x\neq 0$).

1. If $f_{1,2}(1)$, $f_{2,2}(x)$, $f_{1,3}(3)$ form an arithmetic progression, find $x$. 2. Suppose $\left\{f_{0,1}\!\left(\dfrac 1{x_n}\right)\right\}$ ($n\in\mathbb N^{*}$) is a geometric progression with ratio $\dfrac 32$, and $x_1,x_5\in\mathbb N^{*}$. Does there exist a positive integer $u$ with $x_1\geqslant u^4$ and $x_5\leqslant(u+1)^4$? If so, find $u$; if not, explain why. 3. Call a sequence $\{y_n\}$ bounded if there is a positive constant $M$ with $|y_n|\leqslant M$ for all $n$. Let $a>0$, let $m$ be a positive even integer, and let $\{x_n\}$ satisfy $x_1=b<0$ and $x_{n+1}=f_{b,a}\!\left(\dfrac 1{x_n^m}\right)$. Prove that $\{x_n\}$ is bounded if and only if $ab^{m-1}+2\geqslant 0$.