
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 true 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