
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
Does there exist a non-constant polynomial with integer coefficients $Q(x)$ such that for every positive integer $n > 2$, the values $Q(0), Q(1), \cdots, Q(n-1)$ taken modulo $n$ have at most $0.499n$ different remainders?

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