
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 a set $A \subset \mathbb{N}^{*}$, defined as follows: the number of elements in set $A$ is $n$, and $n \geq 2$. If $B=\{z|z=x+y, x \in A, y \in A, x \neq y\}$, then set $B$ is called the derived sum set of set $A$.  
$(Ⅰ)$ When $A_{1}=\{1,2,3,4\}$ and $A_{2}=\{1,2,4,7\}$, write down the derived sum sets of sets $A_{1}$ and $A_{2}$.  
$(Ⅱ)$ When $n=6$, find the maximum and minimum number of elements in the derived sum set $B$ of set $A$.

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