
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
Consider a sequence $\{a_n\}$ whose sum of the first n terms is $S_n$. Given that $a_1=1$ and $S_n=n a_n - n(n-1)$ for $n=1,2,3,\dots$.
1. Find the general term formula for the sequence $\{a_n\}$.
2. If the sum of the first n terms of the sequence $\left\{ \frac {1}{a_n a_{n+1}} \right\}$ is $T_n$, determine the smallest positive integer $n$ such that $T_n > \frac{100}{209}$.

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