
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 \( G = \{(x, y) \mid x \in I, y \in I\} \). 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. \( f(zx, zy) = z^{k} f(x, y) \), where \( k \) is a positive number independent of \( x, y, \) and \( z \).

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