
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
Find all pairs of positive integers  $(a,b)$  satisfying the following conditions:


- $a$  divides  $b^4+1$ ,
- $b$  divides  $a^4+1$ ,
- $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$ .


*Yang Liu*

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