
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 $d$, 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 formula $S\_n$ for the first $n$ terms;

(II) Let $b\_n=\frac{S\_n}{n+k} (k < 0)$, if $\{{b\_n}\}$ is an arithmetic sequence, find the sum $T\_n$ of the first $n$ terms for the sequence $\{\frac{1}{{b\_n}{b\_{m+1}}}\}$.

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