
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
Find the number of triples of sets \((A, B, C)\) such that:
(a) \(A, B, C \subseteq \{1, 2, 3, \ldots, 8\}\).
(b) \(|A \cap B| = |B \cap C| = |C \cap A| = 2\).
(c) \(|A| = |B| = |C| = 4\).

Here, \(|S|\) denotes the number of elements in the set \(S\).

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