
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 $I=[0,1]$ and $G=\{(x, y) \mid x \in I, y \in I\}$. Determine all mappings $f$ from $G$ to $I$ such that for any $x, y, z \in I$, the following conditions hold:
1. $f(f(x, y), z)=f(x, f(y, z))$;
2. $f(x, 1)=x$ and $f(1, y)=y$;
3. $f(z x, z y)=z^{k} f(x, y)$, where $k$ is a positive constant independent of $x, y, z$.

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