
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 its first $n$ terms denoted as $S_n$, and a sequence $\left\{ \frac{b_n}{n} \right\}$ which is an arithmetic sequence with a common difference of $1$. If $a_1=2b_1$, $a_4-a_2=12$, and $S_4+2S_2=3S_3$,
- $(I)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
- $(II)$ Let $c_n= \begin{cases} \frac {n}{b_n(n+2)} & (n \text{ is odd}) \\ \frac {2}{a_n} & (n \text{ is even}) \end{cases}$, and $T_n$ be the sum of the first $n$ terms of $\{c_n\}$, find $T_{2n}$.

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