Problem 1: Let $x, y, z \ge 0$. Prove that $$\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z. \tag{1}$$
Background: I came up with the problem when I tried to prove If $a+b+c=0$ and $\{a,b,c\}\subset[-1,1]$ so $\sum\limits_{cyc}\sqrt{1+a+\frac{7}{9}b^2}\geq3$. Problem 1 can be solved in a similar manner as Prove that $\sum\limits_{cyc}\sqrt{\frac{8ab+8ac+9bc}{(2b+c)(b+2c)}}\geq5$. However, it is a computer solution and very complicated. I want to see if there is nice proofs for problem 1. Also, Problem 1 is verified by Mathematica.
The equality cases: $(x, y, z) = (1,1,1), (1,2,0)$.
Any comments and solutions are welcome and appreciated.
More details: Problem 1 is equivalent to Problem 2 and Problem 3 as follows. Problem 3 implies If $a+b+c=0$ and $\{a,b,c\}\subset[-1,1]$ so $\sum\limits_{cyc}\sqrt{1+a+\frac{7}{9}b^2}\geq3$
Problem 2: Let $x, y, z \ge 0$ with $x + y + z = 3$. Prove that $$\sqrt{x + \frac{7}{9}(y - 1)^2} + \sqrt{y + \frac{7}{9}(z-1)^2} + \sqrt{z + \frac{7}{9}(x-1)^2} \ge 3.$$
Problem 3: Let $a, b, c \ge -1$ with $a + b + c = 0$. Prove that $$\sqrt{1 + a + \frac{7}{9}b^2} + \sqrt{1 + b + \frac{7}{9}c^2} + \sqrt{1 + c + \frac{7}{9}a^2} \ge 3.$$