
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 $f(x)=-3x^2+6x$, let ${S_n}$ be the sum of the first $n$ terms of the sequence ${{a_n}}$. The points $(n, {S_n})$ (where $n \in \mathbb{N}^*$) lie on the curve $y=f(x)$.

(I) Find the general formula for the terms of the sequence ${{a_n}}$.

(II) If ${b_n}={(\frac{1}{2})^{n-1}}$ and ${c_n}=\frac{{a_n} \bullet {b_n}}{6}$, let ${T_n}$ be the sum of the first $n$ terms of the sequence ${c_n}$. Determine whether a maximum value of ${T_n}$ exists. If so, find the maximum value; if not, explain the reason.

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