
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 $\{a_n\}$ with common difference $d \neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\{a_n^2\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\{b_n\}$ with a common ratio $q$ that is a positive rational number less than $1$. The first term of this geometric sequence is $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. Find the possible value(s) of $q$ such that $\frac{S_3}{T_3}$ is a positive integer.

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