
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
In a sequence  $a_1, a_2, . . . , a_{1000}$  consisting of  $1000$  distinct numbers a pair  $(a_i, a_j )$  with  $i < j$  is called *ascending* if  $a_i < a_j$  and *descending* if  $a_i > a_j$  . Determine the largest positive integer  $k$  with the property that every sequence of  $1000$  distinct numbers has at least  $k$  non-overlapping ascending pairs or at least  $k$  non-overlapping descending pairs.

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