
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 $f(x)=\dfrac{x}{x+3}$, and the sequence $\left\{ a_n \right\}$ satisfies $a_1=1$, $a_{n+1}=f(a_n)$ $(n\in \mathbb{N}^*)$.

 (Ⅰ) Find the general formula for the sequence $\left\{ a_n \right\}$, $a_n$;

 (Ⅱ) If $b_n=\dfrac{1}{2}a_na_{n+1}\cdot 3^n$, $S_n=b_1+b_2+\cdots +b_n$, find $S_n$.

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