
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 a common difference $d$ that is an integer, 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 expression for $a\_n$.
2. Let $a\_1 > 1$, the sum of the first $n$ terms of the arithmetic sequence $\{a\_n\}$ be $S\_n$, and the sum of the first $n$ terms of the geometric sequence $\{b\_n\}$ with the first term $1$ and common ratio $q (q > 0)$ 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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