
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
Define $||x||$ $(x\in R)$ as the integer closest to $x$ (when $x$ is the arithmetic mean of two adjacent integers, $||x||$ takes the larger integer). Let $G(x)=||x||$. If $G(\frac{4}{3})=1$, $G(\frac{5}{3})=2$, $G(2)=2$, and $G(2.5)=3$, then $\frac{1}{G(1)}+\frac{1}{G(2)}+\frac{1}{G(3)}+\frac{1}{G(4)}=$______; $\frac{1}{{G(1)}}+\frac{1}{{G(\sqrt{2})}}+\cdots+\frac{1}{{G(\sqrt{2022})}}=$______.

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