
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 list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list  $10,34,34,22,30,22$  the sublist  $22,30,22$  is good and  $10,34,34,22$  is not. A list is very good if all its sublists are good. Find the minimum value of  $k$  such that there exists a very good list of length  $2019$  with  $k$  different values on it.

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