
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
There are 2019 plates arranged in a circle, each with one pastry on it. Petya and Vasya are playing a game. On each turn, Petya points to a pastry and names a number from 1 to 16, and Vasya moves the indicated pastry the specified number of plates either clockwise or counterclockwise (the direction is chosen by Vasya each time). Petya aims to eventually accumulate at least $k$ pastries on one of the plates, while Vasya tries to prevent this from happening. What is the greatest value of $k$ for which Petya can guarantee success?

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