AMO-Bench 2025 Problem 50

Let the polynomial $f(x)=x^6 - 11x^5 + 28x^4 + 2x^3 + 57x^2 + 13x + 30$, and let $\alpha$ be the smallest positive real root of the equation $f(x)=0$. Set $N$ as the remainder of the closest integer to $\alpha^{35}$ divided by 1498. A logistics company warehouses pallets with serial numbers $m=1,2,\dots, M$. To simplify the sampling-judgment'' process, three quality inspectors use their respective cyclic-power residue'' rules: Inspector A uses a cycle of $7$ pallets'' rule to record the remainder of serial number $m$ divided by $7$. If this remainder is a non-zero quadratic residue of modulo $7$ (there exists an $r$ coprime to $7$ such that $r^2\equiv m\pmod7$), it is marked as passed''; Inspector B uses a cycle of $11$ pallets'' rule to record the remainder of serial number $m$ divided by $11$. If this remainder is a non-zero cubic residue of modulo $11$ (there exists an $r$ coprime to $11$ such that $r^3\equiv m\pmod{11}$), it is marked as passed''; Inspector C uses a cycle of $13$ pallets'' rule to record the remainder of serial number $m$ divided by $13$. If this remainder is a non-zero quartic residue of modulo $13$ (there exists an $r$ coprime to $13$ such that $r^4\equiv m\pmod{13}$), it is marked as passed''. A serial number $m$ is marked as qualified'' if and only if all three inspectors mark it as passed''. Let the total number of qualified serial numbers be counted within the range $1 \le m \le M$, where the integer $M$ satisfies $M-724=N$. How many ``qualified'' serial numbers are there in total?