
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
Consider the sequence $\{a_n\}$ with general term ${a_n = n^2\left( \cos^2 \frac{n\pi}{3} - \sin^2 \frac{n\pi}{3} \right)}$. Let ${S_n}$ denote the sum of its first $n$ terms.

(1) Find ${S_n}$;

(2) Define a new sequence $\{b_n\}$ where ${b_n = \frac{S_{3n}}{n \cdot 4^n}}$, and find the sum ${T_n}$ of its first $n$ terms.

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