
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
On the number line, point $A$, $B$ represent numbers $a$, $b$ respectively. Point $C$ lies between $A$ and $B$, representing number $c$.  
$(1)$ If $a=2$, $b=6$, then the distance between $A$ and $B$ is ______;  
$(2)$ If $AC=CB$, then point $C$ is called the midpoint of segment $AB$.  
① If $a=2$, $b=-6$, then $c=$ ______;  
② If $c=-1$, and point $A$ is moved $10$ units to the right to coincide with point $B$, then $a=$ ______;  
③ In general, express $c$ in terms of $a$ and $b$ as $c=$ ______;  
$(3)$ If $AC=\lambda CB$ (where $\lambda\ \ \gt 0$).  
① When $a=-2$, $b=4$, $\lambda=\frac{1}{2}$, then $c=$ ______;  
② In general, express $c$ in terms of $a$, $b$, and $\lambda$ as $c=$ ______.

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