
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 sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{n}$ represents the $n$-th digit after the decimal point of $\sqrt{2}=1.41421356237⋯$ (for example, $a_{1}=4$, $a_{6}=3)$, and $b_{1}=a_{1}$, with ${b_{n+1}}={a_{{b_n}}}$ for all $n\in N^{*}$, find all values of $n$ that satisfy $b_{n}=n-2022$.

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