
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
Petya and Vasya ride bicycles at constant speeds around a circle. Petya's speed is 8 km/h, and Vasya's speed is 10 km/h. Initially, they rode in opposite directions (Petya clockwise and Vasya counterclockwise), and then Petya changed his direction (started moving counterclockwise) and simultaneously doubled his speed. After this, the cyclists met $k$ times less frequently. Find $k$.

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