Divide Set Algorithm to recive the vector (1,1,...,1)

So I'm stuck at this question, the goal is to find a lower bound.
I have a set of natural numbers, each action is
choose a number>1 (refer as k) and divide the whole set (the vector (a1,a2,...,an) by this k then insted of the number's value enter this formula: i/k+i%k (Div+Mod)

for example:
(13,5,8,40) for k=4 we get: (13/4+13%4,5/4+5%4,8/4+8%4,40/4+40%4)
(4,2,2,10) then repeat the action.

-We know the higest number has value N
I need a lower bound of minimum action (If it helps i know the upper bound is 2*sqrt(N))

It's really urgent!!!
Thanks in advance,


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