
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 $A(1,-2,0)$, $B(2,k,-3)$, $C(2,0,2)$, and vector $\overrightarrow{a}=(-3,4,5)$.<br/>$(1)$ If $\overrightarrow{AB}⊥\overrightarrow{a}$, find the value of the real number $k$;<br/>$(2)$ Find the cosine value of the angle between vectors $\overrightarrow{AC}$ and $\overrightarrow{a}$.

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