
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 vectors \( \boldsymbol{a} \), \( \boldsymbol{b} \), and \( \boldsymbol{c} \) satisfying
$$
|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{c}|=1: k: 3 \quad \left(k \in \mathbf{Z}_{+}\right),
$$
and \( \boldsymbol{b} - \boldsymbol{a} = 2(\boldsymbol{c} - \boldsymbol{b}) \),
if \( \alpha \) is the angle between \( \boldsymbol{a} \) and \( \boldsymbol{c} \), then \( \cos \alpha \) is ________.

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