
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  $S$  be the set of positive real numbers. Find all functions  $f\colon S^3 \to S$  such that, for all positive real numbers  $x$ ,  $y$ ,  $z$  and  $k$ , the following three conditions are satisfied:

(a)  $xf(x,y,z) = zf(z,y,x)$ ,

(b)  $f(x, ky, k^2z) = kf(x,y,z)$ ,

(c)  $f(1, k, k+1) = k+1$ .

(*United Kingdom*)

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