
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
A math interest group at a certain middle school encountered some new mathematical symbols during an extracurricular study and exploration. They excerpted some materials as follows: For three real numbers $a$, $b$, $c$, $M\{a,b,c\}$ represents the average of these three numbers, and $\min \{a,b,c\}$ represents the smallest of these three numbers. For example: $M\{1,2,9\}=\frac{1+2+9}{3}=4, \min\{1,2,-3\}=-3$. Please solve the following problems based on the above materials:
$(1) M\{{2}^{2},\sqrt{9},-{3}^{2}\}=$______；
$(2)$ If $M\{-2x$,$x^{2}$,$3\}=2$, find the value of $x$;
$(3)$ If $a \gt 0$, and the point $P(M\{-2,a-1,2a\}$,$\min \{-2,a-1,2a\})$ is on the graph of the inverse proportion function $y=\frac{-2}{x}$, find the value of $a$.

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