
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 that the function $f(x)$ is defined on the domain $[0,+\infty)$ and satisfies $f(x)=2f(x+2)$, when $x\in[0,2)$, $f(x)=-2x^{2}+4x$. Let the maximum value of $f(x)$ on $[2n-2,2n)$ be $a_{n}(n\in\mathbb{N^{+}})$, and the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, then $S_{n}=$____.

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