
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 arithmetic sequence ${a_n}$ has a common difference of a positive number, $a_1=1$, and the sum of its first $n$ terms is $S_n$. The geometric sequence ${b_n}$ has $b_1=2$, and it satisfies the equations $b_2S_2=12$ and $b_2+S_3=10$.
(I) Find the general term formulas for the sequences ${a_n}$ and ${b_n}$.
(II) Let $c_n=b_n+\frac{1}{S_n}$. Find the sum of the first $n$ terms, $T_n$, of the sequence ${c_n}$.

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