
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
For any set  $S$ , let  $P(S)$  be its power set, the set of all of its subsets. Over all sets  $A$  of  $2015$  arbitrary finite sets, let  $N$  be the maximum possible number of ordered pairs  $(S,T)$  such that  $S \in P(A), T \in P(P(A))$ ,  $S \in T$ , and  $S \subseteq T$ . (Note that by convention, a set may never contain itself.) Find the remainder when  $N$  is divided by  $1000.$ *Proposed by Ashwin Sah*

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