
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
A railway shuttle operates between points $A$ and $B$, such that the train returns to $A$ immediately after arriving at $B$. Alongside the railway, there is also a cableway. A mine cart departs from $A$ at the same time as the train. After half an hour, the mine cart is as far from $A$ as the train is from $B$. After the mine cart travels an additional 2 km, the positions of the two vehicles become symmetric with respect to the midpoint of the $AB$ line segment (meanwhile, the train has turned back from $B$). After a quarter-hour from this point, the two vehicles meet for the first time. What is the distance between $A$ and $B$ and the speeds of the two vehicles? (Assume that the speeds are constant.)

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