AMO-Bench 2025 Problem 49

In the Cartesian plane, there is a lamp at every integer lattice point, each with two states: on'' and off''. The lamp board features a special switching system: pressing any lamp toggles the states of all lamps in its entire row and entire column (on to off, and off to on). Define $f(x)=x^2-18x+23$ and $g(x)=37x+18$. The equation $f(f(x))=g(x)$ has real roots $a_1,a_2,a_3,a_4$ with $a_1 < a_2 < a_3 < a_4$. You can operate the switches at coordinates $(x,y)$ where $1\leq x\leq 17$ and $1\leq y\leq a_2+a_3$. Regardless of the initial state of the lamp board, what is the minimum number of lamps within this operable rectangular region that you can guarantee will be lit after performing operations using this system?