
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
Let $S\_n$ be the sum of the first $n$ terms of the sequence ${a\_n}$, and $2a\_n + S\_n = An^2 + Bn + C$.

1. Find $a\_n$ when $A=B=0$, $C=1$.
2. If the sequence ${a\_n}$ is an arithmetic sequence and $A=1$, $C=-2$.
   a. Find $a\_n$.
   b. Let $b\_n = 2^n a\_n$, and let $T\_n$ be the sum of the first $n$ terms of the sequence ${b\_n}$. For any positive integer $n$, $T\_n - (2n + m) \cdot 2^{n+1} > 0$ always holds. Find the range of values for the real number $m$.

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