
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
Two cars, referred to as car A and car B, set off from locations $A$ and $B$ respectively, heading towards each other (car B departs from $B$). The speed of car B is $k$ times the speed of car A $(k > 1)$. Upon reaching their destinations $B$ and $A$, both cars immediately return. On the return trip, car A maintains its original speed, while car B’s speed changes to match car A's speed. During their return, the two cars meet at point $C$. If $D$ is the midpoint of the route from $A$ to $B$, and $\frac{C D}{A D}=\frac{1}{2}$, find the value of $k$. (Please provide the solution process)

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