
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 $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ that satisfy the following conditions:
(i) \( f(x y z) + f(x) + f(y) + f(z) = f(\sqrt{x y}) f(\sqrt{y z}) f(\sqrt{z x}) \) for all \( x, y, z \in \mathbb{R}^{+} \).
(ii) \( f(x) < f(y) \) for all \( 1 \leq x < y \).

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