
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 the function $f(x)=2x+1$, and the sequence $\{a_n\}$ satisfies $a_n=f(n)$ where $n\in\mathbb{N}^*$, the sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and $b_1=2$, $T_n=b_{n+1}-2$ for $n\in\mathbb{N}$.  
(1) Find the general formulas for $\{a_n\}$ and $\{b_n\}$.  
(2) Define $x=[x]+(x)$, where $[x]$ is the integer part of the real number $x$, and $(x)$ is the fractional part, with $0\leqslant (x) < 1$. Let $c_n=\left( \frac{a_n}{b_n}\right)$, find the sum of the first $n$ terms of the sequence $\{c_n\}$.

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