
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
The positive sequence  $\{a_n\}$  satisfies: $a_1=1+\sqrt 2$  and  $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of  $\{a_n\}$ ;
(2)Find the set of all the positive integers  $n$  so that  $\lfloor a_n\rfloor=2022$ .

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