
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 two curves $f\left(x\right)=x^{3}+ax$ and $g\left(x\right)=x^{2}+bx+c$ both pass through the point $P\left(1,2\right)$, and have a common tangent line at point $P$.  
$(1)$ Find the values of $a$, $b$, and $c$;  
$(2)$ Let a moving point $M$ on the parabola $g\left(x\right)=x^{2}+bx+c$ have a distance $d$ to the line $y=3x-2$. Find the minimum value of $d$.

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