
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
Let $\{a_{n}\}$ be a sequence of positive terms, and let $S_{n}$ denote the sum of the first $n$ terms of the sequence. It is known that ${a}_{1}=1$ and $2({S}_{n}+{S}_{n-1})={a}_{n}^{2}+1$ for $n\geq 2$.  
$(1)$ Find $a_{n}$;  
$(2)$ Define ${b}_{n}=\left\{\begin{array}{l}\frac{{a}_{n}+1}{2}, \text{if } n \text{ is odd}\\ \frac{4}{{a}_{n}({a}_{n}+4)}, \text{if } n \text{ is even}\end{array}\right.$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find the value of $T_{20}$.

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