polynomials – FFT: multiplication of multiple poynomials in time O (KSlogS)

I have a problem where I have to use the K polynomials of the K algorithm of the fast Fourier Transform (FFT), …, PK where

degree (P1) + · · · + grade (PK) = S

I must show that I can find the product of these K polynomials in O (SlogSlogK).

I know that the FFT execution time is O (nlogn). So far I only find the solution in time O (KSlogS). The suggestions say you should use the trees and the divide and conquer method, but I'm still so blank.

Any advice and help are very appreciated! Thank you.