
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
Define: For three distinct negative integers, if the arithmetic square root of the product of any two of them is an integer, then these three numbers are called "combination square numbers." For example, for the numbers $-1$, $-4$, $-9$, the square roots of the products $(-1) \times (-4)$, $(-1) \times (-9)$, and $(-4) \times (-9)$ are $2$, $3$, and $6$, respectively, which are all integers. Therefore, $-1$, $-4$, $-9$ are called "combination square numbers." <br/> $(1)$ Are $-4$, $-16$, $-25$ "combination square numbers"? Please explain the reason; <br/> $(2)$ If the three numbers are $-3$, $m$, $-12$ and they are "combination square numbers", where the arithmetic square root of the product of two of them is $12$, find the value of $m$; <br/> $(3)$ Write a set of "combination square numbers" containing $-2$.

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