
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 that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_{n}=n^{2}+n+1$. In the positive geometric sequence ${b_n}$, $b_3=a_2$, $b_4=a_4$. Find:
1. The general term formulas for ${a_n}$ and ${b_n}$;
2. If $c_n$ is defined as $c_n=\begin{cases} a_{n},(n\text{ is odd}) \\ b_{n},(n\text{ is even}) \end{cases}$, and $T_n=c_1+c_2+…+c_n$, find $T_{10}$.

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