
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 sequence $\{a_{n}\}\left(n\in N*\right)$ with the first term $a_{1}=1$ and the sum of the first $n$ terms denoted as $S_{n}$. Let $\lambda$ and $k$ be constants. If for all positive integers $n$, the equation $S_{n+1}^{\frac{1}{k}}-S_n^{\frac{1}{k}}=\lambda a_{n+1}^{\frac{1}{k}}$ holds, then the sequence is called a "$\lambda -k$" sequence.<br/>$(1)$ If an arithmetic sequence $\{a_{n}\}$ is a "$\lambda -1$" sequence, find the value of $\lambda$;<br/>$(2)$ If the sequence $\{a_{n}\}$ is a "$\frac{\sqrt{3}}{3}-2$" sequence, and $a_{n} \gt 0$, find the general formula for the sequence $\{a_{n}\}$;<br/>$(3)$ For a given $\lambda$, determine if there exist three different sequences $\{a_{n}\}$ that form a "$\lambda -3$" sequence, where $a_{n}\geqslant 0$. If they exist, find the range of values for $\lambda$; if not, explain the reason.

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