
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 a sequence $\{a_n\}$ of positive terms, the sum of the first $n$ terms is $S_n$, where $S_n= \frac {1}{4} a_n^{2}+ \frac {1}{2}a_n$ ($n\in \mathbb{N}^*$);  
(1) Find $a_n$;  
(2) Define a new sequence $\{b_n\}$ as follows:  
$$
b_n= 
\begin{cases}
a_n, & \text{if } n \text{ is odd} \\
b_{\frac{n}{2}}, & \text{if } n \text{ is even}
\end{cases}
$$
and let $c_n = b_{2^n+4}$ ($n \in \mathbb{N}^*$). Find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$;  
(3) If $b_n=\lambda q^{a_n}+\lambda$ ($\lambda$, $q$ are constants, $q>0$ and $q\neq 1$), and $c_n = 3+n+(b_1+b_2+\dots+b_n)$, does there exist a pair of real numbers $(\lambda, q)$ such that the sequence $\{c_n\}$ forms a geometric progression? If such a pair exists, find the pair $(\lambda, q)$ and the general term of the sequence $\{c_n\}$. If not, explain why.

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