
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 positive term sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, $a_1=1$, and $(t+1)S_n = a_n^2 + 3a_n + 2$ for $t\in \mathbb{R}$.

$(1)$ Find the general term formula for the sequence $\{a_n\}$;

$(2)$ Suppose sequence $\{b_n\}$ satisfies $b_1=1$, $b_{n+1} - b_n = a_{n+1}$, find the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{2b_n + 7n} \right\}$, denoted as $T_n$.

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