
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 any set  $S$  of integers, let  $f(S)$  denote the number of integers  $k$  with  $0 \le k < 2019$  such that there exist  $s_1, s_2 \in S$  satisfying  $s_1 - s_2 = k$ . For any positive integer  $m$ , let  $x_m$  be the minimum possible value of  $f(S_1) + \dots + f(S_m)$  where  $S_1, \dots, S_m$  are nonempty sets partitioning the positive integers. Let  $M$  be the minimum of  $x_1, x_2, \dots$ , and let  $N$  be the number of positive integers  $m$  such that  $x_m = M$ . Compute  $100M + N$ .

*Proposed by Ankan Bhattacharya*

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