
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 $S\_n$, where the first term $a\_1 \neq 0$ and $2S\_n = a\_n^2 + a\_n$ for all $n \in \mathbb{N}^+$.

1. Find the general formula for the $n$-th term of the sequence $\{a\_n\}$.
2. If $a\_n > 0$, let $b\_n = \frac{4}{a\_n(a\_n + 2)}$. The sum of the first $n$ terms of the sequence $\{b\_n\}$ is denoted as $T\_n$. If $T\_n < m$ always holds true, where $m \in \mathbb{Z}$, find the minimum value of $m$.

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