
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  $n=2017$  and  $x_1,\dots,x_n$  be boolean variables. An \emph{ $7$ -CNF clause} is an expression of the form  $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$ , where  $\phi_1,\dots,\phi_7$  are each either the function  $f(x)=x$  or  $f(x)=1-x$ , and  $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$ . For example,  $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12}$  is a  $7$ -CNF clause. What's the smallest number  $k$  for which there exist  $7$ -CNF clauses  $f_1,\dots,f_k$  such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is  $0$  for all values of  $(x_1,\dots,x_n)\in\{0,1\}^n$ ?

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