
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(x)$  be a polynomial with the following properties:
(i)   $ f(0)\equal{}0$ ; (ii)  $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$  is an integer for any two different integers  $ a$  and  $ b$ . Is there a polynomial which has these properties, although not all of its coefficients are integers?

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