
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 the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$.

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