
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
Consider two sequences: an arithmetic sequence ${a_{n}}$ with sum $S_{n}$ of its first $n$ terms, and a geometric sequence ${b_{n}}$. Given that $a_{1}=3$, $b_{1}=1$, $b_{2}+S_{2}=10$, and $a_{5}-2b_{2}=a_{3}$.

(I) Find the general term formulas for both sequences ${a_{n}}$ and ${b_{n}}$.
(II) Define a new sequence ${c_{n}=\frac{1}{2S_{n}}+b_{n}}$. Let $T_{n}$ denote 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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