
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
Determine the functions $f$ that satisfy the following conditions:

(i) $f$ is defined on the set of integers;

(ii) $f(z)$ is a rational number for every integer $z$;

(iii) if $f(x)<c<f(y)$ and $c$ is rational, then $f$ takes the value $c$; and

(iv) if $x, y, z$ are integers and their sum is zero, then

\[ 
f(x) + f(y) + f(z) = f(x)f(y)f(z)
\]

holds.

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