
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 $\{a_n\}$ be an arithmetic sequence, and $\{b_n\}$ be a geometric sequence, with $b_1=a_1^2$, $b_2=a_2^2$, $b_3=a_3^2$ ($a_1<a_2$), and $$\lim_{n\rightarrow +\infty }(b_{1}+b_{2}+\cdots +b_{n})= \sqrt {2}+1.$$ Find the first term and the common difference of $\{a_n\}$.

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