
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 $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$ and $a_1=1$, and it is known that $S_{n+1} = S_n + a_n + n + 1 \left(n \in \mathbb{N}^* \right)$.

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

(2) Let $T_n$ denote the sum of the first $n$ terms of the sequence $\left\{\frac{1}{a_n}\right\}$, find the smallest positive integer $n$ that satisfies the inequality $T_n \geqslant \frac{19}{10}$.

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