
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 the sequence $\{{a_n}\}$ with the sum of its first $n$ terms ${S_n} = {n^2} + 2n$,

(I) Find the general formula for the $n$th term ${a_n}$;

(II) Let $2{b_n} = {a_n} - 1$, and ${T_n} = \frac{1}{{{b_1}{b_2}}} + \frac{1}{{{b_2}{b_3}}} + \frac{1}{{{b_3}{b_4}}} + \cdots + \frac{1}{{{b_n}{b_{n + 1}}}}$. Find ${T_n}$.

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