Take b = 2.

b^(n-1) mod n = 1.

353 is prime.

b^((n-1)/353)-1 mod n = 344337, which is a unit, inverse 558310.

11 is prime.

b^((n-1)/11)-1 mod n = 659772, which is a unit, inverse 498113.

(11 * 353) divides n-1.

(11 * 353)^2 > n.

n is prime by Pocklington's theorem.