
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
For a pair of integers  $a$  and  $b$ , with  $0 < a < b < 1000$ , set  $S\subseteq \{ 1, 2, \dots , 2003\}$  is called a *skipping set* for  $(a, b)$  if for any pair of elements  $s_1, s_2 \in S$ ,  $|s_1 - s_2|\not\in \{ a, b\}$ . Let  $f(a, b)$  be the maximum size of a skipping set for  $(a, b)$ . Determine the maximum and minimum values of  $f$ .

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