
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 $\{a\_n\}$ where all terms are positive, the sum of the first $n$ terms is $S\_n = \frac{1}{2}(a\_n - 1)(a\_n + 2)$, where $n \in \mathbb{N}^*$.

1. Find the general formula for the $n$-th term of the sequence $\{a\_n\}$.
2. Let $b\_n = (-1)^n a\_n a_{n+1}$, find the sum $T_{2n}$ of the first $2n$ terms of the sequence $\{b\_n\}$.

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