
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 that the arithmetic sequence $\{a\_n\}$ has a common difference of $d$, and the geometric sequence $\{b\_n\}$ has a common ratio of $q$. Let $S\_n$ and $T\_n$ be the sum of the first $n$ terms of $\{a\_n\}$ and $\{b\_n\}$, respectively. If $n^2(T\_n + 1) = 2^n S\_n$, for all positive integers $n$, find the values of $d$ and $q$.

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