
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 the sum of the first $n$ terms as $S_{n}$, where the common difference $d$ is an integer, $S_{3}=21$, and $a_{1}$, $a_{2}+1$, $a_{7}$ form a geometric sequence.<br/>$(Ⅰ)$ Find the general formula for $\{a_{n}\}$;<br/>$(Ⅱ)$ Find the sum $T_{n}$ of the first $n$ terms of the sequence $\{\frac{5}{a_{n}a_{n+1}}\}$.

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