
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 a new operation: For any rational numbers $a$ and $b$, we have $a\triangle b=-2ab-b+1$, where the right side of the equation represents the usual addition, subtraction, and multiplication operations. For example, $2\triangle 5=-2\times 2\times 5-5+1=-20-5+1=-25+1=-24$.  
$(1)$ Find the value of $3\triangle 7$;  
$(2)$ If $a=3$ and $b=-4$, calculate the results of $a\triangle b$ and $b\triangle a$, and determine whether the commutative law applies to the "$\triangle$" operation.

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