Question 16
Find the maximum value of $k$ such that: for all $a, b, c$ are sidelengths of some triangle, then
$$
k \cdot \frac{3(a^2 + b^2 + c^2)}{(a + b + c)^2} + \sqrt{\frac{ab + bc + ca}{a^2 + b^2 + c^2}} \leq k + 1.
$$
$$
k \cdot \frac{3(a^2 + b^2 + c^2)}{(a + b + c)^2} + \sqrt{\frac{ab + bc + ca}{a^2 + b^2 + c^2}} \leq k + 1.
$$