
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 $a_{2}=3a_{1}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence. It is known that ${S_{n+1}}+{S_n}+{S_{n-1}}=3{n^2}+2$ for all $n\geq 2, n\in \mathbb{N}^{*}$. If for any $n\in \mathbb{N}^{*}$, $a_{n} \lt a_{n+1}$, then the range of possible values for the first term $a_{1}$ is ______.

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