
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
Find all pairs of integers \((c, d)\) with \(c, d > 1\) that have the following property: For any degree \(d\) monic polynomial \(Q\) with integer coefficients, and any prime \(p\) such that \(p > c(2c + 1)\), there exists an integer set \(S\) with at most \(\frac{2c - 1}{2c + 1} p\) elements such that the set
\[
\bigcup_{s \in S} \{s, Q(s), Q(Q(s)), \ldots\}
\]
contains a complete residue system modulo \(p\).

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