2020 AIME II Problem 14

For real number x let \lfloor x\rfloor be the greatest integer less than or equal to x, and define \{x\} = x - \lfloor x \rfloor to be the fractional part of x. For example, \{3\} = 0 and \{4.56\} = 0.56. Define f(x)=x\{x\}, and let N be the number of real-valued solutions to the equation f(f(f(x)))=17 for 0\leq x\leq 2020. Find the remainder when N is divided by 1000.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.