
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 $|A \cap B| = |B \cap C| = |C \cap A| = 1$, $A \cap B \cap C = \emptyset$, and each set excludes one specific common element, e.g., the number 8, from their elements. Given that each set is a subset of $\{1,2,3,4,5,6,7,8\}$ (but excluding the number 8 from being an element of any set), let $M$ be the number of such minimally intersecting ordered triples $(A, B, C)$. Find the remainder when $M$ is divided by $1000$.

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