
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 positive real numbers \(x\) and \(y\) that satisfy \(xy = 1\), find the range of the function
\[ f(x, y) = \frac{x + y}{\lfloor x \rfloor \lfloor y \rfloor + \lfloor x \rfloor + \lfloor y \rfloor + 1} \]
(where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\)).

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