Issues with Cake Eating Problem

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A while back I asked about simulating the cake eating problem in sage and the algorithm seems like an excellent way to work with things. So far the code looks as follows for obtaining a soloution to the cake eating problem:

#Step1: Name variables
k,k1,v0,beta,vk,vk1=var('k,k1,v0,beta,vk,vk1')

#Step2: Initialize your values
beta=0.6
v0=log(k)

#Step 3: The Loop to Obtain Policy function
for n in range(5):
    vk = log(k-k1) + beta*v0(k=k1)
    FOC = vk.diff(k1)
    k1star = solve(FOC==0, k1)
    print(n, k1star)
    v0 = (vk).subs(k1=k1star[0].rhs())

The results I get are as follows:

0 [k1 == 3/8*k]
1 [k1 == 24/49*k]
2 [k1 == 147/272*k]
3 [k1 == 816/1441*k]
4 [k1 == 4323/7448*k]

Initially I thought that it would be a matter of just setting k to some value however this breaks the loop.

If I wanted to obtain values for k1 (and then plot them) how would I go about doing this?