
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
For a sequence of positive terms $\{a_n\}$, the sum of the first $n$ terms $S_n$ satisfies: $S_n^2 - (n^2 + n - 1)S_n - (n^2 + n) = 0$.
1. Find the general formula for the sequence $\{a_n\}$.
2. If the sequence $\{b_n\}$ satisfies $(n+2)^2 \cdot b_n = \frac{n+1}{a_n^2}$, and the sum of the first $n$ terms is $T_n$, and for all $n \in \mathbb{N}^*$, $T_n < \frac{m^2}{5}$ ($m \in \mathbb{R}$), find the range of values for $m$.

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