
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 specifically intersecting if $|A \cap B| = 2$, $|B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Each set is a subset of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $M$ be the number of such specifically 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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