Consider the 3 × 3 world shown below. 80% of the time the agent goes in the direction it selects; the rest of the time it moves at right angles to the intended direction.

r -1 +10
-1 -1 -1
-1 -1 -1

Implement value iteration for this world for each value of r below. Use discounted rewards with a discount factor of 0.99.
Show the policy obtained in each case. Explain intuitively why the value of r leads to each policy.

a) r = 100
b) r = −3
c) r = 0
d) r = +3

The value of sin(x) (in radians) can be approximated by the alternating infinite series create a function (prob3_2) that takes inputs of a scalar angle measure (in radians) and the number of approximation terms, n, and estimates sin(x). do not use the sin function in your solution. you may use the factorial function. though this can be done without a loop (more efficiently), your program must use (at least) one. you may find the mod() function useful in solving the problem.