
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 an arithmetic sequence $\left\{a_{n}\right\}$ with common difference $d \neq 0$, and a geometric sequence $\left\{b_{n}\right\}$ with common ratio $q$ which is a positive rational number less than 1. If $a_{1}=d$, $b_{1}=d^{2}$, and $\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{b_{1}+b_{2}+b_{3}}$ is a positive integer, then find the value of $q$.

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