
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
Let  $k$  be a positive real number. Determine all functions  $f:[-k, k]\rightarrow[0, k]$  satisfying the equation  $$ f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2 $$  for any  $x, y\in[-k, k]$  such that  $x+y\in[-k, k]$ .

*Proposed by Maximiliano Sánchez*

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