
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
Reading material: Like $(\sqrt{5}+2)\times (\sqrt{5}-2)=1$, $\sqrt{a}•\sqrt{a}=a\left(a\geqslant 0\right)$, etc. When two algebraic expressions containing square roots are multiplied, the product does not contain square roots. We call these two expressions rationalizing factors of each other. When performing square root operations, rationalizing factors can be used to eliminate square roots in the denominator. In a math class, the teacher gave a problem: "Given $a=\frac{1}{\sqrt{2}-1}$, find the value of $3a^{2}-6a-1$." Smart student Xiao Ming, based on the above material, provided the following solution: Since $a=\frac{1}{\sqrt{2}-1}=\frac{\sqrt{2}+1}{(\sqrt{2}-1)×(\sqrt{2}+1)}=\sqrt{2}+1$, then $a-1=\sqrt{2}$, so $\left(a-1\right)^{2}=2$, hence $a^{2}-2a+1=2$, so $a^{2}-2a=1$, so $3a^{2}-6a=3$, so $3a^{2}-6a-1=2$. Please solve the following problems based on the above material and Xiao Ming's solution process: <br/>$(1)$ The rationalizing factor of $\sqrt{3}-\sqrt{2}$ is ______, $\frac{1}{\sqrt{3}-\sqrt{2}}=$______; The rationalizing factor of $\sqrt{3}-2$ is ______, $\frac{1}{\sqrt{3}-2}=$______; <br/>$(2)$ If $a=\frac{2}{3-\sqrt{7}}$, find the value of $-2a^{2}+12a+3$.

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