
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
Soldier enjoys studying mathematical problems. After learning about linear equations, he came up with a new definition: if $x_{0}$ is a solution to the linear equation $ax+b=0\left(a\neq 0\right)$ with respect to $x$, and $y_{0}$ is one of the solutions to an equation with respect to $y$, and $x_{0}$ and $y_{0}$ satisfy $x_{0}+y_{0}=100$, then the equation with respect to $y$ is called the "friendly equation" of the linear equation with respect to $x". For example, the solution to the linear equation $3x-2x-99=0$ is $x_{0}=99$, and the solutions to the equation $y^{2}+1=2$ are $y=1$ or $y=-1$. When $y_{0}=1$, $x_{0}+y_{0}=100$, so $y^{2}+1=2$ is the "friendly equation" of the linear equation $3x-2x-99=0$. <br/>$(1)$ Given the equations with respect to $y$: ① $2y-2=4$, ② $|y|=2$, which of the following equations is the "friendly equation" of the linear equation $3x-2x-102=0"? Please write down the correct number directly as ______. <br/>$(2)$ If the equation with respect to $y$ $|2y-2|+3=5$ is the "friendly equation" of the linear equation $x-\frac{2x-2a}{3}=a+1$, find the value of $a$. <br/>$(3)$ If the equation with respect to $y$ $2m|y-49|+\frac{m(y-1)}{45}=m+n$ is the "friendly equation" of the linear equation $mx+45n=54m$, write down the value of $\frac{m+n}{n}$ directly.

Present the answer in LaTex format: \boxed{Your answer}
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