
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  $p = 2027$  be the smallest prime greater than  $2018$ , and let  $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let  $\mathrm{GF}(p)$  be the integers modulo  $p$ , and let  $\mathrm{GF}(p)(X)$  be the set of rational functions with coefficients in  $\mathrm{GF}(p)$  (so that all coefficients are taken modulo  $p$ ). That is,  $\mathrm{GF}(p)(X)$  is the set of fractions  $\frac{P(X)}{Q(X)}$  of polynomials with coefficients in  $\mathrm{GF}(p)$ , where  $Q(X)$  is not the zero polynomial. Let  $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$  be a function satisfying \[
    D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2}
\]for any  $f,g\in \mathrm{GF}(p)(X)$  with  $g\neq 0$ , and such that for any nonconstant polynomial  $f$ ,  $D(f)$  is a polynomial with degree less than that of  $f$ . If the number of possible values of  $D(P(X))$  can be written as  $a^b$ , where  $a$ ,  $b$  are positive integers with  $a$  minimized, compute  $ab$ .

*Proposed by Brandon Wang*

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