
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 $A(1,2)$, $B(-1,4)$, $C(4,k)$.  
$(1)$ If $A$, $B$, $C$ are collinear points, find $|\overrightarrow{AC}|$.  
$(2)$ If $\overrightarrow{AB}⊥(\overrightarrow{AB}+\overrightarrow{AC})$, find $cos〈\overrightarrow{AB},\overrightarrow{AC}〉$.

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