
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)  $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$ .
(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}
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