Claims About a Cubic Iteration

A sequence $\{a_n\}$ satisfies $a_{n+1}=\dfrac 14(a_n-6)^3+6$ ($n\in\mathbb N^{*}$). Determine, with proof, which of the following claims are true.

1. When $a_1=3$, $\{a_n\}$ is decreasing, and there is a constant $M\leqslant 0$ with $a_n>M$ for all $n$. 2. When $a_1=5$, $\{a_n\}$ is increasing, and there is a constant $M\leqslant 6$ with $a_n<M$ for all $n$. 3. When $a_1=7$, $\{a_n\}$ is decreasing, and there is a constant $M>6$ with $a_n>M$ for all $n$. 4. When $a_1=9$, $\{a_n\}$ is increasing, and there is a constant $M>0$ with $a_n<M$ for all $n$.