
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
Car A and car B start from points A and B simultaneously and move towards each other (car B starts from point B). The speed of car B is \( k \) times the speed of car A (\( k > 1 \)). After reaching points B and A, respectively, they immediately return. On the return journey, car A maintains its original speed, but car B's speed equals that of car A. During the return journey, the two cars meet at point C. If the midpoint of AB is point D and \(\frac{CD}{AD} = \frac{1}{2}\), find \( k \). (Please show the solution process)

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