
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
For the sequence $\{a_n\}$ with each term being a positive number and the sum of the first $n$ terms being $S_n$, it satisfies: $S_n = \frac{1}{4}a_n^2 + \frac{1}{2}a_n + \frac{1}{4}$ $(n \in \mathbb{N}^*)$.

(1) Find $a_n$.

(2) Suppose the sequence $\{a_n + b_n\}$ is a geometric sequence with the first term being $1$ and the common ratio $q$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$.

For the advanced class students:

The sequence $\{a_n\}$ with each term being a positive number and the sum of the first $n$ terms being $S_n$, satisfies: $S_n = \frac{1}{4}a_n^2 + \frac{1}{2}a_n + \frac{1}{4}$ $(n \in \mathbb{N}^*)$.

(1) Find $a_n$.

(2) Let the function $f(n) = \begin{cases} a_n, & \text{if } n \text{ is odd} \\ f\left( \frac{n}{2} \right), & \text{if } n \text{ is even} \end{cases}$. Let $C_n = f(2^n + 4)$ $(n \in \mathbb{N}^*)$, 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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