
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 bus travels with a constant speed between points $A$ and $B$ starting at 6 AM. Upon arriving at either point $A$ or $B$, the bus turns back immediately. Petya on a moped and Vasya on a bicycle set off together with the bus from point $A$ towards point $B$ at 6 AM, each traveling at a constant speed. It is known that during the second trip of the bus from point $A$ to point $B$, it caught up with Vasya at 1:30 PM and arrived at point $B$ at 3 PM simultaneously with Petya. Determine the ratio of Vasya's speed to Petya's speed. Also, specify the time when the bus first caught up with Vasya on its way from point $B$ to point $A$.

(12 points)

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