
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 $S_{n}$, satisfying $2S_n=n^2+n$ ($n\in\mathbb{N}^*$), and let ${b_n}=(-1)^n\frac{a_{2n+1}}{a_n \cdot a_{n+1}}$, then the sum of the first 2020 terms of the sequence $\{b_{n}\}$, $T_{2020}=$____.

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