
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  $f = f(x_1, x_2, \cdots, x_n)$  only equals  $0$  or  $1$ , then define  $f$  as an " $n$ -variable Boolean function" and denote   $$ D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \} $$  . $(1)$  Determine the number of  $n$ -variable Boolean functions;  $(2)$  Let  $g$  be a  $10$ -variable Boolean function satisfying   $$ g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2} $$  Evaluate the size of the set  $D_{10} (g)$  and  $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$ .  

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