
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
Let $f(x)=(x-a)(x-b)(x-c)$, where real numbers $a$, $b$, and $c$ satisfy $2b=a+c$.  
$(1)$ If $b=0$ and $c=1$, find the equation of the tangent line to the function $f(x)$ at $x=1$.  
$(2)$ If $b-a=3$, find the extreme values of the function $f(x)$.  
$(3)$ If the curve $f(x)$ has three distinct common points with the line $y=-(x-b)-2$, find the range of $c-a$.

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