
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 $\{a\_n\}$, it satisfies that the points $(a\_n, a\_{n+1}) (n \in \mathbb{N}^*)$ are on the graph of the function $y = 2x$, and $a\_1 = 1$.

1. Find the general term formula of the sequence $\{a\_n\}$.
2. Let $b\_n = \frac{3}{(\log\_2 a\_{n+1}) \cdot (\log\_2 a\_{n+2})}$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$.
3. Let the function $f(x) = -x^2 + 2ax - a^2 + a - 1$ ($a$ is a constant), and $T\_n > f(x)$ holds for any $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, find the range of the real number $a$.

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