
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 and answer the questions: For the average of three numbers $a$, $b$, $c$, the minimum number can be represented by symbols. We define $M\{a,b,c\}$ as the average of $a$, $b$, $c$, $\min \{a,b,c\}$ as the minimum of $a$, $b$, $c$, and $\max \{a,b,c\}$ as the maximum of $a$, $b$, $c$. For example: $M\{-1,2,3\}=\frac{-1+2+3}{3}=\frac{4}{3}$, $\min \{-1,2,3\}=-1$, $\max \{-1,2,3\}=3$; $M\{-1,2,a\}=\frac{-1+2+a}{3}=\frac{a+1}{3}$, $\min \{-1,2,a\}=\left\{\begin{array}{l}{a(a≤-1)}\\{a(a>-1)}\end{array}\right.$<br/>$(1)$ Please fill in the blanks: $\max \{-2,3,c\}=\_\_\_\_\_\_$; if $m \lt 0$, $n \gt 0$, $\max \{3m,\left(n+3\right)m,-mn\}=$______<br/>$(2)$ If $\min \{2,2x+2,4-2x\}=2$, find the range of $x$;<br/>$(3)$ If $M\{2,x+1,2x\}=\min \{2,x+1,2x\}$, find the value of $x$.

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