
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 the sequence $\{a\_n\}$, it is known that $a_{n+1} = a_n + 4$ and $a_1 + a_4 = 14$.
(1) Find the general term formula $a_n$ and the formula for the sum of the first $n$ terms $S_n$.
(2) Let $b_n = \frac{S_n}{n+k}$ ($k \in \mathbb{Z}$). If $\{b_n\}$ is an arithmetic sequence, and the sum of the first $n$ terms of the sequence $\{\frac{1}{b_n b_{n+1}}\}$, denoted as $T_n$, is always less than or equal to $\frac{m}{100}$, find the minimum value of the positive integer $m$.

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