
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 $\{a_n\}$, the sum of its first $n$ terms is $S_n$, $a_1=1$, and $S_{n+1}=S_n+a_n+n+1$ for $n\in\mathbb{N^*}$.  
$(1)$ Find the general term formula for the sequence $\{a_n\}$.  
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n} \right\}$ be $T_n$. 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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