
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  $S = \{1,2,\cdots,2013\}$ . Let  $N$  denote the number  $9$ -tuples of sets  $(S_1, S_2, \dots, S_9)$  such that  $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$  for  $n=1,2,3,4$ . Find the remainder when  $N$  is divided by  $1000$ .

*Proposed by Lewis Chen*

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