
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
Let  $a_1$ ,  $a_2$ ,  $\ldots\,$ ,  $a_{2019}$  be a sequence of real numbers.  For every five indices  $i$ ,  $j$ ,  $k$ ,  $\ell$ , and  $m$  from 1 through 2019, at least two of the numbers  $a_i$ ,  $a_j$ ,  $a_k$ ,  $a_\ell$ , and  $a_m$  have the same absolute value.  What is the greatest possible number of distinct real numbers in the given sequence?

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