
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
Decide whether or not there exists a nonconstant polynomial \( Q(x) \) with integer coefficients with the following property: for every positive integer \( n > 2 \), the numbers
\[ Q(0), Q(1), Q(2), \ldots, Q(n-1) \]
produce at most \( 0.499n \) distinct residues when taken modulo \( n \).

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