
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 $f(x) = 2f(x+2)$, defined on $[0,+\infty)$, where $f(x) = -2x^2 + 4x$ when $x \in [0, 2)$, determine the range of $S_n$, where $S_n$ is the sum of the first $n$ terms of the sequence ${a_n}$, with $a_n$ denoting the maximum value of $f(x)$ on $[2n-2, 2n)$.

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