
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 \( f \) be a function defined on the set of nonnegative real numbers and taking values within this set. Find all functions \( f \) that satisfy the following conditions:
(i) \( f[x \cdot f(y)] \cdot f(y) = f(x+y) \);
(ii) \( f(2) = 0 \);
(iii) \( f(x) \neq 0 \) when \( 0 \leqslant x < 2 \).

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