
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  $F$  be the family of all sets of positive integers with  $2010$  elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let  $f$  be a function defined from  $F$  to the positive integers such that  $f(K)$  is the biggest element of  $K \in F$ . Determine the least value of  $f(K)$ .

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