
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
The sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted as $S_n$, with $a_1=t$. The point $(S_n, a_{n+1})$ lies on the line $y=2x+1$, where $n\in\mathbb{N}^*$.  
(1) If the sequence $\{a_n\}$ is a geometric sequence, find the value of the real number $t$;  
(2) Let $b_n=n\cdot a_n$, under the condition of (1), find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$;  
(3) For a sequence $\{c_n\}$ where each term is non-zero, the number of integers $i$ satisfying $c_i\cdot c_{i+1}<0$ is called the "sign-changing count" of the sequence $\{c_n\}$. Let $$c_{n}= \frac {b_{n}-4}{b_{n}}$$ (where $n\in\mathbb{N}^*$), under the condition of (2), find the "sign-changing count" of the sequence $\{c_n\}$.

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