(a) You set up a (3, 37) Shamir threshold scheme, working modulo the prime 227. Three of the shares are (1,4), (2,8), and (3, 16). Another share is (5,x), but the part denoted by x is unreadable. Find the correct value of x, the relevant polynomial, and the message. Justify all your steps. (b) In a (4,41) Shamir threshold scheme working modulo the prime 229, the shares (1,9), (2, 27), (3,81), and (4,243) were given to Alice, Bob, Jerry, and Charles. Calculate the corresponding Lagrange interpolation polynomial p(x) modulo 229; that is, write p(x) = a + ax + ax + a3x with ao, a1, A2, A3 Z229. Also, identify the secret. (c) Verify the solutions of Parts (a) and (b) in sagemath.