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I remember reading this problem in some contest, but I don't remember where. I think the contest was 2016 or so. I searched on a few websites with no luck. Do you happen to know what contest it was?
The problem is: Given integers a1, ..., an, We define x0 = 1 and for every 
. Find xn. The solution was O(n * polylog(n)) with FFT.
Is it Kyoya and Train's solution?
I remember 2 such problems from CSacademy: Colored Forests and Jetpack. Both are basically some easy combinatorics + what you're asking for.
Jetpack was the one I'm looking for, thanks!
I wrote this task on Codechef as well (late 2014): task / solution (It is like O(n log^2 n))
I saw you wrote a very detailed editorial. It must have taken lots of time.
Compliments.