
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 $\left\{ a_n \right\}$ with a common difference $d \ne 0$, it is known that $a_1 + a_4 = 14$, and $a_1, a_2, a_7$ form a geometric sequence.

(Ⅰ) Find the general term formula $a_n$ and the sum of the first $n$ terms formula $S_n$;

(Ⅱ) Let $b_n = \frac{S_n}{n+k} \left( k \ne 0 \right)$, if $\left\{ b_n \right\}$ is an arithmetic sequence, find the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n b_{n+1}} \right\}$, denoted as $T_n$.

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