
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 us consider a set  $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$ , satisfying the following properties:  $f(a_i) < 2003$  and  $f(a_i) = f(a_j) \quad \forall i, j$  from  $\{1, 2,\ldots , 2004\}$ , where  $f(a_i)$  denotes number of elements which are relatively prime with  $a_i$ . Find the least positive integer  $k$  for which in every  $k$ -subset of  $S$ , having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

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