
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
Determine all pairs \((a, b)\) of real numbers such that

$$
a \cdot \lfloor b \cdot n \rfloor = b \cdot \lfloor a \cdot n \rfloor
$$

holds for all positive integers \(n\).

(For a real number \(x\), \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).)

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