
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
Read the following material: For three numbers $a$, $b$, $c$, let $M\{a,b,c\}$ represent the average of these three numbers, and let $\min \{a,b,c\}$ represent the smallest number among these three numbers. For example: $M\{-1,2,3\}=\frac{-1+2+3}{3}=\frac{4}{3}$; $\min \{-1,2,3\}=-1$; $\min \{-1,2,a\}=\left\{\begin{array}{l}a(a≤-1)\\-1(a>-1)\end{array}\right.$ Solve the following problems: 

$(1)$ Fill in the blank: If $\min \{2,2x+2,4-2x\}=2$, then the range of $x$ is ______.

$(2)$ If $M\{2,x+1,2x\}=\min \{2,x+1,2x\}$, find $x$.

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