
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  $a,$  which satisfy the following condition:

For every sequence   $a_1,a_2,a_3,\ldots$  of pairwise different positive integers, for which the inequality  $a_n\leq an$  holds for every positive integer  $n,$  there exist infinitely many numbers in the sequence with sum of their digits in base  $4038,$  which is not divisible by  $2019.$ 

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