
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
1. Let \( x_i \in \{0,1\} \) (for \( i=1,2,\cdots,n \)). If the value of the function \( f=f(x_1, x_2, \cdots, x_n) \) only takes 0 or 1, then \( f \) is called an n-ary Boolean function. We denote
   \[
   D_{n}(f)=\left\{(x_1, x_2, \cdots, x_n) \mid f(x_1, x_2, \cdots, x_n)=0\right\}.
   \]
   (1) Find the number of n-ary Boolean functions.

   (2) Let \( g \) be a 10-ary Boolean function such that
   \[
   g(x_1, x_2, \cdots, x_{10}) \equiv 1+\sum_{i=1}^{10} \prod_{j=1}^{i} x_{j} \ (\bmod \ 2),
   \]
   find the number of elements in the set \( D_{10}(g) \), and evaluate
   \[
   \sum_{\left(x_1, x_2, \cdots, x_{10}\right) \in D_{10}(g)}\left(x_1+x_2+\cdots+x_{10}\right).
   \]

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