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As you learned in your Data Structures and Algorithms course, a recursive function is a function that computes a value by making a call to itself. This implies that a recursive function uses previous values to compute new ones. To prevent an infinite loop, a recursive function terminates when a base case is reached.

The general template for a recursive function is:

First, evaluate the base case(s) of the function. These serve as the termination point for the recursion.
If the base cases are not reached, perform necessary operations and then call the function again.
Return the computed value.

The most common example of this is the Fibonacci sequence. It is written in Python as follows:

def fib(n):
  if n < 2:
    return n
  k = fib(n - 1) + fib(n - 2)
  return k         
# Alternatively, you can simply return fib(n - 1) + fib(n - 2) without assigning it to a variable

Instructions

Complete the Python function factorial(n) that recursively computes n!. Remember the following identities:

0! = 1
Factorials are not defined for numbers less than zero; your function should return 1 for such inputs.

A modified Fibonacci sequence is a sequence of numbers in which each number is the sum of the preceding three numbers (as opposed to two). Complete the Python function mod_fib(n) that computes the n^th number in this sequence.

Use these initial conditions:

F(0) = 0
F(1) = 1
F(2) = 2

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