pythonKelly has a magic pot which can slowly cook one ingredient into others. She already

mastered the cooking techniques and concludes that the process can be modeled using

markov chain represented by a transition matrix $A.$

You are given the ingredient list material$\_$list $=["$snake oil", "silver",

"caviar", "bones", "uranium", "plutonium"$]$ which corresponds to the rows and

columns order of A$.$ Each entry $A_{ij}$ of A represents the fraction of the $j^{th}$ ingredient of

material$\_$list that will turn into the $i^{th}$ ingredient per minute without losing any mass.

One day Kelly started making some of her magical soup and after a few minutes she

realized she forgot to start the timer. She could not remember the time she started

cooking, or even the ingredient composition of her initial soup. She can only recall that

at the beginning of the cooking, $(25+-0\mathrm{.}1)\%$ of the mass of all materials is snake oil.

In fear of ruining her precious ingredients, she immediately casts a spell to determine

the current composition of the soup and stores it in the vector comp, where each entry

is the fraction in terms of mass of each ingredient, following the order in

material$\_$list.

Write a code snippet to help Kelly with the following:

Use the current ingredient composition given in comp to determine the initial

ingredient composition. Store in the variable initial$\_$frac the initial fraction of

the ingredient plutonium.

Suppose Kelly decides to leave the soup cooking for a very long time, until the

ingredient composition in the pot is no longer changing. Help her determine this

final ingredient composition and store it in the vector final$\_$comp, where each

entry is the fraction in terms of mass of each ingredient, following the order in

material$\_$list.

The setup code will provide the following variables:

Your code snippet should define the following variables: