
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 $f(x)$ satisfies the equation $f(x) + f(1-x) = \frac{1}{2}$ for any $x \in \mathbb{R}$.

(1) The sequence ${a_n}$ is defined as: $a_n = f(0) + f(\frac{1}{n}) + f(\frac{2}{n}) + \dots + f(\frac{n-1}{n}) + f(1)$. Find $a_n$.

(2) Let $b_n = \frac{4}{4a_n - 1}$, $T_n = b_1^2 + b_2^2 + b_3^2 + \dots + b_n^2$, and $S_n = 32 - \frac{16}{n}$. Compare the size of $T_n$ and $S_n$.

(3) Under the conditions of (1), let $b_n = 4a_n - 1$ and $c_n = b_nq^{n-1} (q \neq 0, n \in \mathbb{N}^*)$. Find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.

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