
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
The sequence $\{a_n\}$, where each term is positive, has a sum of the first $n$ terms denoted by $S_n$, and it satisfies: $S_n= \frac {1}{4}a_n^{2}+ \frac {1}{2}a_n+ \frac {1}{4}$ for $n\in\mathbb{N}^{+}$.  
(I) Find $a_n$;  
(II) Let the function $f(n)= \begin{cases} a_n,& \text{if } n \text{ is odd} \\ f\left( \frac {n}{2}\right),& \text{if } n \text{ is even} \end{cases}$, and $C_n=f(2^{n}+4)$ for $n\in\mathbb{N}^{+}$, find the sum of the first $n$ terms of the sequence $\{C_n\}$, denoted by $T_n$.

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