
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 $A$ be a non-empty subset of the set of positive integers. The set $B=\{|u-v||u,v\in A, u\neq v\}$ is called the generating set of set $A$.  
$(1)$ When $A=\{1,3,6\}$, write down the generating set $B$ of set $A$;  
$(2)$ If $A$ is a set consisting of 5 positive integers, find the minimum number of elements in its generating set $B$;  
$(3)$ Determine whether there exists a set $A$ consisting of 4 positive integers such that its generating set $B=\{2,3,5,6,10,16\}$, and explain the reason.

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