
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 non-zero common difference $d$, where $a_1 + a_4 = 14$, and $a_1, a_2, a_7$ form a geometric sequence.
(1) Determine the general term formula $a_n$ and the sum of the first $n$ terms formula $S_n$ of the sequence $\{a_n\}$;
(2) Define $b_n = \frac{S_n}{n+k}$. If $\{b_n\}$ is an arithmetic sequence, find the sum of the first $n$ terms $T_n$ of the sequence $\left\{\frac{1}{b_n b_{n+1}}\right\}$.

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