
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 geometric sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and the common ratio $q > 0$, it is known that $S_2=2a_2-2$ and $S_3=a_4-2$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) Let $c_n= \begin{cases} \frac {\log _{2}a_{n}}{n^{2}(n+2)}, & \text{for odd } n \\ \frac {n}{a_{n}}, & \text{for even } n\end{cases}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{c_n\}$. Find $T_{2n}$.

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