
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Exploration:<br/>Material 1: When solving the equation system $\left\{{\begin{array}{l}{({a-1})+2({b+2})=6}\\{2({a-1})+({b+2})=6}\end{array}}\right.$, a "substitution method" was used, and the solution is as follows:<br/>Solution: Let $a-1=x$, $b+2=y$, the original system of equations can be transformed into $\left\{{\begin{array}{l}{x+2y=6}\\{2x+y=6}\end{array}}\right.$,<br/>Solving, we get $\left\{{\begin{array}{l}{x=2}\\{y=2}\end{array}}\right.$, which means $\left\{{\begin{array}{l}{a-1=2}\\{b+2=2}\end{array}}\right.$, leading to $\left\{{\begin{array}{l}{a=3}\\{b=0}\end{array}}\right.$.<br/>Material 2: When solving the equation system $\left\{{\begin{array}{l}{4x+10y=6①}\\{8x+22y=10②}\end{array}}\right.$, a "global substitution" method was used, and the solution is as follows:<br/>Solution: Transforming equation ② $8x+20y+2y=10$ into $2\left(4x+10y\right)+2y=10$③, substituting equation ① into ③, we get $2\times 6+2y=10$, which leads to $y=-1$; substituting $y=-1$ into equation ①, we get $x=4$, so the solution to the system of equations is $\left\{{\begin{array}{l}{x=4}\\{y=-1}\end{array}}\right.$.<br/>Based on the above materials, solve the following problems:<br/>$(1)$ Using the substitution method to solve for the solution of the system of equations in terms of $a$ and $b$: $\left\{{\begin{array}{l}{({\frac{a}{4}-1})+2({\frac{b}{3}+2})=4}\\{2({\frac{a}{4}-1})+({\frac{b}{3}+2})=5}\end{array}}\right.$;<br/>$(2)$ If the solution to the system of equations in terms of $x$ and $y$ is $\left\{{\begin{array}{l}{x=10}\\{y=6}\end{array}}\right.$, find the solution to the system of equations in terms of $m$ and $n$: $\left\{{\begin{array}{l}{5{a_1}({m-3})+3{b_1}({n+2})={c_1}}\\{5{a_2}({m-3})+3{b_2}({n+2})={c_2}}\end{array}}\right.$;<br/>$(3)$ Given $x$, $y$, $z$ satisfy $\left\{{\begin{array}{l}{3x-2z+12y=47①}\\{2x+z+8y=36②}\end{array}}\right.$, find the value of $z$.

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