
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 points $A(1,1)$, $B(3,-1)$, and $C(a,b)$:  
$(1)$ If points $A$, $B$, and $C$ are collinear, find the relationship between $a$ and $b$;  
$(2)$ If $\overrightarrow{AC} = 2\overrightarrow{AB}$, find the coordinates of point $C$.

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