
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 $\left\{a_{n}\right\}$ be an arithmetic sequence, and $\left\{b_{n}\right\}$ be a geometric sequence, with $b_{1}=a_{1}^{2}, b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}$ $\left(a_{1}<a_{2}\right)$. Additionally, it is given that $\lim _{n \rightarrow+\infty}\left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$. Find the first term and the common difference of the sequence $\left\{a_{n}\right\}$.

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