
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 $\{a_n\}$, where $a_1=1$ and the sum of the first $n$ terms is $S_n$. If for any $n\in \mathbb{N}^*$, it holds that $S_n=a_{n+k}-k$ (where $k$ is a constant and $k\in \mathbb{N}^*$), then the sequence $\{a_n\}$ is called an "$H(k)$ sequence".

(1) If the sequence $\{a_n\}$ is an "$H(1)$ sequence", find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$.

(2) If the sequence $\{a_n\}$ is an "$H(2)$ sequence" and $a_2$ is an integer, is there a sequence $\{a_n\}$ such that $|a_n^2-a_{n-1}a_{n+1}|\leqslant 40$ holds for any $n\geqslant 2$, $n\in \mathbb{N}^*$? If it exists, find all possible values of $a_2$ for such sequences; if not, please explain why.

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