
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 $S_{n}=\{A|A=(a_{1},a_{2},\ldots,a_{n})\}$, where $a_{i}=0$ or $1$, for $i=0, 1, 2, \ldots, n\}$, for $U, V\in S_{n}$, $d(U,V)$ represents the number of elements in $U$ and $V$ that are different.  
$(1)$ Let $U=\{1,1,1,1,1,1\}$, there exist $m$ elements $V\in S_{6}$, such that $d(U,V)=2$, then $m=$ _______ ;  
$(2)$ If a certain $U\in S_{n}$ is given, for any $V\in S_{n}$, then the sum of all $d(U,V)$ is _______ .

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