
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
Define an ordered triple  $ (A, B, C)$  of sets to be minimally intersecting if  $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$  and  $ A \cap B \cap C \equal{} \emptyset$ . For example,  $ (\{1,2\},\{2,3\},\{1,3,4\})$  is a minimally intersecting triple. Let  $ N$  be the number of minimally intersecting ordered triples of sets for which each set is a subset of  $ \{1,2,3,4,5,6,7\}$ . Find the remainder when  $ N$  is divided by  $ 1000$ .

**Note**:  $ |S|$  represents the number of elements in the set  $ S$ .

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