
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
Find all real numbers \(a\) (where \(a \geq 1\)) that satisfy the following condition: For every sequence \(\{a_1, a_2, \ldots\}\) composed of distinct positive integers, if for any positive integer \(n\), \(a_n \leq an\), then there exist infinitely many terms in the sequence such that their digit sums in base 4038 are not divisible by 2019.

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