
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 \(x_{i} \in \{0,1\} (i=1,2, \cdots, n)\). If the function \(f=f(x_1, x_2, \cdots, x_n)\) can only take values 0 or 1, then \(f\) is called an \(n\)-variable Boolean function. Define
\[ D_n(f) = \{(x_1, x_2, \cdots, x_n) \mid f(x_1, x_2, \cdots, x_n) = 0\}. \]

(1) Find the number of \(n\)-variable Boolean functions;

(2) Let \(g\) be an \(n\)-variable Boolean function such that
\[ g(x_1, x_2, \cdots, x_n) \equiv 1 + \sum_{i=1}^{n} \prod_{j=1}^{i} x_{j} \ (\bmod \ 2), \]

Find the number of elements in the set \(D_n(g)\) and determine the largest positive integer \(n\) such that
\[ \sum_{(x_1, x_2, \cdots, x_n) \in D_n(g)} (x_1 + x_2 + \cdots + x_n) \leq 2017. \]

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