Value function interation help

Hi,

I have a task to solve a utility optimization problem with dynamic programming using value function interation.

The task is quite similar to the classic cake eating problem, which there exist codes on, but I cant seem to work it out. Here are the information:

Utility function:

subject to:

You can choose the variables freely if you want, but in the problem Beta = 0.96, n=2 and alpha = 1/3

All help would be appriciated.