
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
For an arithmetic sequence $\{a_{n}\}$, the first term is $a_{1}=1$, and the sum of the first $10$ terms is $S_{10}=100$. In a sequence of positive terms in geometric progression $\{b_{n}\}$, $b_{1}=2a_{1}$, and $b_{3}=a_{4}+1$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Find the sum of the first $2n+1$ terms of the sequence $\{\left(-1\right)^{n-1}(a_{n}+1)^{2}\}$.  
$(3)$ Given $c_{n}=\left\{\begin{array}{l}{{a_n}{b_n},\text{ for odd } n}\\{\frac{{({3{a_n}-4}){b_n}}}{{{a_n}{a_{n+2}}}},\text{ for even } n}\end{array}\right.$, find the sum of the first $2n$ terms of the sequence $\{c_{n}\}$.

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