
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
Given the following definition: We call the ordered real number pair $\left(a,b,c\right)$ the subsidiary coefficient pair of the quadratic polynomial $ax^{2}+bx+c$ with respect to $x$, and we call the quadratic polynomial $ax^{2}+bx+c$ the subsidiary polynomial of the ordered real number pair $\left(a,b,c\right)$.  
$(1)$ The subsidiary coefficient pair of the quadratic polynomial $3x^{2}+2x-1$ with respect to $x$ is ______;  
$(2)$ The difference between the subsidiary polynomial of the ordered real number pair $\left(2,a,1\right)$ and the subsidiary polynomial of the ordered real number pair $\left(1,-2,4\right)$ does not contain a linear term. Find the value of $a$.

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