
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
Let  $S=\{1,2,\cdots, n\}$  and let  $T$  be the set of all ordered triples of subsets of  $S$ , say  $(A_1, A_2, A_3)$ , such that  $A_1\cup A_2\cup A_3=S$ . Determine, in terms of  $n$ ,
\[ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|\]

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