
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 $\{a_n\}$ whose sum of the first n terms $S_n$ satisfies $S_1 > 1$, and $6S_n = (a_n+1)(a_n+2)$ for $n \in \mathbb{N}^{*}$,

(1) Find the relationship between $a_n$ and $a_{n+1}$.
(2) Among all the sequences $\{a_n\}$ that satisfy the condition, find the minimum value of $a_{2015}$.
(3) If every term of the sequence $\{a_n\}$ is positive, let another sequence $\{b_n\}$ satisfy $a_n(2^{b_n}-1)=3$, and denote $T_n$ as the sum of the first n terms of $\{b_n\}$. Is there a constant c such that $T_n \geq c$ holds for any positive integer n? If it exists, state the range of values for c; if not, explain why.

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