
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 positive sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, for any $n \in \mathbb{N}^*$, the point $(a_n, S_n)$ lies on the graph of the function $f(x) = \frac{1}{2}x^2 + \frac{1}{2}x$.
1. Find the first term $a_1$ and the general formula for $a_n$;
2. If the sequence $\{b_n\}$ satisfies $\log_2 b_n = n + \log_2(2a_n - 1) (n \in \mathbb{N}^*)$, find the sum of the first $n$ terms of $\{b_n\}$, $T_n$;
3. Given that the sequence $\{c_n\}$ satisfies $c_n = \frac{4n - 6}{T_n - 6} - \frac{1}{a_n a_{n+1}} (n \in \mathbb{N}^*)$. If for any $n \in \mathbb{N}^*$, there exists $x_0 \in [-\frac{1}{2}, \frac{1}{2}]$ such that $c_1 + c_2 + \ldots + c_n \leqslant f(x) - a$ holds, find the range of the real number $a$.

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