
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 an integer common difference $d$, and $a_k = k^2 + 2$, $a_{2k} = (k + 2)^2$, where $k$ is a constant and $k \in \mathbb{N}^+$,
(1) Find the value of $k$ and the general term $a_n$.
(2) Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ be $S_n$, and consider a geometric sequence $\{b_n\}$ with the first term $l$ and common ratio $q$ (where $q > 0$), and the sum of the first $n$ terms of $\{b_n\}$ be $T_n$. If there exists a positive integer $m$ such that $\frac{S_2}{S_m} = T_3$, find the value of $q$.

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