$4.$ Create a Python$($or Pseudo Python$)$ function that solves how many non$-$negative integer solutions there are in a linear equation in the form:

$\backslash [$

x$\_\{1\}+$x$\_\{2\}+$x$\_\{3\}+$x$\_\{4\}+\backslash$ldots$+$x$\_\{\backslash$ell$\}=$S

$\backslash ]$

Where each $\backslash ($ x$\_\{$i$\}\backslash )$ variable may have some restriction on its value. The less than or equal to constraints are stored in a list lower$\_$constraints $(0$ is stored if there is no constraint$).$ The upper bound constraints are stored in the list upper$\_$constraints as PAIRS: $\backslash (($i$,$ k$)\backslash )$ indicating that $\backslash ($ x$\_\{$i$\}\backslash$leq k $\backslash ),$ and len$($upper$\_$constraints$)\backslash (\backslash$leq $3\backslash ).[$Hint: Be careful of double constraints $($eg$.\backslash (3\backslash$leq x$\_\{2\}\backslash$leq $8\backslash ))]$

Write a program that determines the number of non$-$negative integer solutions. Your program should accept as input the variables LENGTH $\backslash (=1\backslash )$ and VALUE $\backslash (=$S $\backslash )($these should be defined constants at the top of your code$).$

Your code should do the following:

$-$ Check the number of upper bound constraints, then appropriately compute the PIE for that number of constraints $($hint: you should have conditionals where you then have the formula for the $2-$set and $3-$set PIE$)$

$-$ If one of your computations of setting aside bars yields a negative number of stars to place, you should ensure your code computes this to be $0.$

$-$ The lower constraints should be applied first and continue throughout your code.

$-$ Your code must be WELL COMMENTED $-$ your algorithmic structure should be very clear. If needed, it is recommended you also include a paragraph describing your code in your assignment essentially a README

$-$ Your code must be less than $60$ lines.