
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 being $S_{n}=n^{2}+r$, where $r$ is a constant.<br/>$(1)$ Find the value of $r$;<br/>$(2)$ Let ${b_n}=\frac{1}{2}({a_n}+1)$, determine 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}
user