新定义:若一元一次方程的解在一元一次不等式组解集范围内,则称该一元一次方程为该不等式组的"关联方程",例如:方程$x-1=3$的解为$x=4$,而不等式组$\left\{\begin{array}{l}{x-1＞1}\\{x-2＜3}\end{array}\right.$的解集为$2 \lt x \lt 5$,不难发现$x=4$在$2 \lt x \lt 5$的范围内,所以方程$x-1=3$是不等式组$\left\{\begin{array}{l}{x-1＞1}\\{x-2＜3}\end{array}\right.$的"关联方程".
$(1)$在方程①$3\left(x+1\right)-x=9$；②$4x-7=0$；③$\frac{x-1}{2}+1=x$中,不等式组$\left\{\begin{array}{l}{2x-2＞x-1}\\{3(x-2)-x≤4}\end{array}\right.$的"关联方程"是______$;(填序号)$
$(2)$若关于$x$的方程$2x-k=6$是不等式组$\left\{\begin{array}{l}{\frac{3x+1}{2}≥x}\\{\frac{x-1}{2}≥\frac{2x+1}{3}-2}\end{array}\right.$的"关联方程",求$k$的取值范围?