
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = \frac{n^2}{2} + \frac{3n}{2}$. If the sequence $\{b_n\}$ satisfies $b_n = a_{n+2} - a_n + \frac{1}{a_{n+2} \cdot a_n}$, then the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$, equals \_\_\_\_\_\_.

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