
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  $n$  to be a positive integer. Given a set  $\{ a_1, a_2, \ldots, a_n \} $  of integers, where  $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$   $\forall i$ , we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to  $0$ . If all of these sums have different remainders when divided by  $2^n$ , we say that  $\{ a_1, a_2, \ldots, a_n \} $  is *$n$ -complete*.

For each  $n$ , find the number of *$n$ -complete* sets.

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