
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
Given the set
$$
\begin{array}{c}
A_{k}=\left\{2^{k-1}+\sum_{i=0}^{k-2} a_{i} 2^{a_{i}+i} \mid a_{i} \in\{0,1\},\right. \\
i=0,1, \cdots, k-2\} .
\end{array}
$$

where \(n_{k}\) represents the sum of all elements in the set \(A_{k}\), find the value of 
$$
\sum_{k=1}^{2015} n_{k}=
$$

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