
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
Find the smallest positive integer  $c$  for which the following statement holds: Let  $k$  and  $n$  be positive integers. Suppose there exist pairwise distinct subsets  $S_1$ ,  $S_2$ ,  $\dots$ ,  $S_{2k}$  of  $\{1, 2, \dots, n\}$ , such that  $S_i \cap S_j \neq \varnothing$  and  $S_i \cap S_{j+k} \neq \varnothing$  for all  $1 \le i,j \le k$ . Then  $1000k \le c \cdot 2^n$ .

*Proposed by Yang Liu*

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