
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
Find all  $k>0$  such that there exists a function  $f : [0,1]\times[0,1] \to [0,1]$  satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$ ; $f(x,y) = f(y,x)$ ; $f(x,1)=x$ ; $f(zx,zy) = z^{k}f(x,y)$ , for any  $x,y,z \in [0,1]$ 

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