
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
(1) Let ${S_n}$ be the sum of the first $n$ terms of the sequence ${{a_n}}$, where ${{a_n} > 0}$, and ${S_n= \frac{1}{6}{a_n}\left( {{a_n}+3 \right)}}$. Find the general term of the sequence ${{a_n}}$.

(2) The sequence ${a_n}$ satisfies ${a_1=0}$ and ${a_{n+1}= \frac{{a_n}- \sqrt{3}}{\sqrt{3}{a_n}+1}}$ (${n \in \mathbb{N}^*}$). Find ${a_{2015}}$.

(3) In the sequence ${{a_n}}$, ${{a_n}= \frac{1}{n+1}+ \frac{2}{n+1}+ \cdots + \frac{n}{n+1}}$ and ${{b_n}= \frac{2}{{{a_n}{a_{n+1}}}}}$. Find the sum of the first $n$ terms of the sequence ${{b_n}}$, denoted as ${S_n}$.

(4) The solution of the equation ${f\left( x \right)=x}$ is called the fixed point of the function ${f\left( x \right)}$. If ${f\left( x \right)= \frac{{ax}}{x+2}}$ has a unique fixed point and the sequence ${\left\{ a_n \right\}}$ satisfies ${a_1= \frac{1}{2}}$ and ${\frac{1}{a_{n+1}}=f\left( \frac{1}{a_n} \right)}$, find ${a_{2018}}$.

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