
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 sets $A,B$ be subsets of the real number set $R$, and denote: $A+B=\{a+b|a\in A,b\in B\}$.

$(1)$ Given $A=\{0,1,2\}$, $B=\{-1,3\}$, try to represent $A+B$ using the enumeration method;

$(2)$ Let $a_1= \frac{2}{3}$, when $n\in{N}^{*}$, and $n\geqslant 2$, the focal length of the curve $\frac{x^2}{n^2-n+1}+ \frac{y^2}{1-n}= \frac{1}{9}$ is $a_n$, if $A=\{a_1,a_2,\cdots,a_n\}$, $B=\left\{- \frac{1}{9},- \frac{2}{9},- \frac{2}{3}\right\}$, the sum of all elements in $A+B$ is $S_n$. For any positive integers $m,n,k$ satisfying $m+n=3k$, and $m\neq n$, the inequality $S_m+S_n-\lambda S_k > 0$ always holds, find the maximum value of the real number $\lambda$;

$(3)$ If the integer set $A_1\subseteq A_1+A_1$, then $A_1$ is called a "self-generating set"; if any positive integer is the sum of all elements in some non-empty finite subset of the integer set $A_2$ that does not contain zero, then $A_2$ is called a "basis set of $N^*$". The question is: does there exist an integer set that is both a self-generating set and a basis set of $N^*$? Please explain your reasoning.

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