
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 \( S = \{1, 2, 3, 4, \ldots, 100000\} \). Find the least possible value of \( k \) such that any subset \( A \) of \( S \) with \( |A| = 2010 \) contains two distinct numbers \( a \) and \( b \) with \( |a - b| \leq k \).

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