
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 $\log_2b_n=n+\log_2(2a_n-1) \, (n\in\mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$;

(3) Given that the sequence $\{c_n\}$ satisfies $c_n= \frac{4n-6}{T_n-6}- \frac{1}{a_na_{n+1}} \, (n\in\mathbb{N}^*)$. If for any $n\in\mathbb{N}^*$, there exists $x_0\in\left[- \frac{1}{2}, \frac{1}{2}\right]$ 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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