
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
For all real numbers  $x$ , we denote by  $\lfloor x \rfloor$  the largest integer that does not exceed  $x$ . Find all functions  $f$  that are defined on the set of all real numbers, take real values, and satisfy the equality
\[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\]
for all real numbers  $x$  and  $y$ .

*Navneel Singhal, India*

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