
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 the vectors \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) such that \( |\boldsymbol{a}| : |\boldsymbol{b}| : |\boldsymbol{c}| = 1 : k : 3 \) where \( k \in \mathbf{Z}_{+} \), and \(\boldsymbol{b} - \boldsymbol{a} = 2(\boldsymbol{c} - \boldsymbol{b}) \). If \(\alpha\) is the angle between \(\boldsymbol{a}\) and \(\boldsymbol{c}\), then \(\cos \alpha = \)

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