
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 came up with ten quadratic trinomials. Then Vasya sequentially named consecutive natural numbers (starting from a certain number), and Petya substituted each named number into one of the trinomials of his choice and wrote the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in that exact order). What is the maximum number of numbers that Vasya could name?

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