Please provide a step-by-step explanation that demonstrates how this all works. Please build up a matrix equation, then calculate the adjugated matrix of a relevant matrix mod 227, followed by the calculation of a relevant determinant, then please illustrate the computation of a relevant inverse mod 227.

This question is about secret sharing.

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 the 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) = a₀ + a₁x + a₂x² + a₃x³ with a₀; a₁; a₂; a₃ Z₂₂₉. Also, identify the secret.

**Please display the results of the polynomial mod 229 computation.