
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
Given a set $\{1, 2, 3, 4, 5, 6, 7, 8\}$, define an ordered triple $(A, B, C)$ of sets as minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Additionally, ensure that each set must contain an exact total of 4 elements each. Calculate $N$, the number of such minimally intersecting ordered triples, and find the remainder when $N$ is divided by $1000$.

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