
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]

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 Determine the least integer  $n$  such that for any set of  $n$  lines in the 2D plane, there exists either a subset of  $1001$  lines that are all parallel, or a subset of  $1001$  lines that are pairwise nonparallel.

*Proposed by Samuel Wang*

<details><summary>Solution</summary>*Solution.*  $\boxed{1000001}$ Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most  $1001$  groups of lines, all pairwise intersecting, with each group containing at most  $1001$  lines. Thus,  $n = 1000^2 + 1 =
\boxed{1000001}$ .</details>

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