
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
Given \( I=[0,1] \) and \( G=\{(x, y) \mid x \in I, y \in I\} \), find 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. \( f(z x, z y)=z^{k} \cdot f(x, y) \), where \( k \) is a positive number that is independent of \( x, y, \) and \( z \).

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