
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 three points in space \\(A(-1,2,1)\\), \\(B(0,1,-2)\\), and \\(C(-3,0,2)\\):
\\((1)\\) Find the cosine of the angle between vectors \\( \overrightarrow{AB} \\) and \\( \overrightarrow{AC} \\).
\\((2)\\) If the vector \\(3 \overrightarrow{AB} - \overrightarrow{AC} \\) is perpendicular to the vector \\( \overrightarrow{AB} + k \overrightarrow{AC} \\), find the value of the real number \\(k\\).

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