
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  $A = (a_1, a_2, \ldots, a_{2001})$  be a sequence of positive integers. Let  $m$  be the number of 3-element subsequences  $(a_i,a_j,a_k)$  with  $1 \leq i < j < k \leq 2001$ , such that  $a_j = a_i + 1$  and  $a_k = a_j + 1$ .  Considering all such sequences  $A$ , find the greatest value of  $m$ .

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