
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 $A$ and $B$ be subsets of the real number set $\mathbb{R}$. Define $A+B=\{a+b \mid a \in A, b \in B\}$.
1. Given $A=\{0,1,2\}$ and $B=\{-1,3\}$, enumerate the elements of $A+B$.
2. Let $a_1= \frac{2}{3}$. For $n \in \mathbb{N}^{*}$ and $n \geq 2$, the focal length of the curve $\frac{x^2}{n^2-n+1} + \frac{y^2}{1-n} = \frac{1}{9}$ is given by $a_n$. If $A=\{a_1,a_2,\dots,a_n\}$ and $B=\{-\frac{1}{9}, -\frac{2}{9}, -\frac{2}{3}\}$, let the sum of all elements in $A+B$ be $S_n$. Find the value of $S_n$.
3. Under the conditions of (2), for any positive integers $m$, $n$, and $k$ that satisfy $m+n=3k$ and $m \neq n$, the inequality $S_m + S_n - \lambda S_k > 0$ always holds. Find the maximum value of the real number $\lambda$.

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