OSN 2022 Problem 8

Determine the smallest positive real $K$ such that the inequality $$K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}}$$holds for any real numbers $0 \le a,b,c \le 1$ .

Proposed by Fajar Yuliawan, Indonesia