One, (20 points) Given real numbers $a, b, c$ satisfy the inequalities
$$
|a| \geqslant|b+c|,|b| \geqslant|c+a|,|r| \geqslant|a+b| .
$$

Find the value of $a+1+r$.


Given $|a| \geqslant|b+c|$, we get $a^{2} \geqslant b^{2}+2 b c+c^{2}$.
Similarly, $b^{2} \geqslant c^{2}+2 c a+a^{2}$, $c^{2} \geqslant a^{2}+2 a b+b^{2}$.
Adding the three equations, we get
$a^{2}+b^{2}+c^{2} \geqslant 2 a^{2}+2 b^{2}+2 c^{2}+2 a b+2 b c+2 c a$,
which simplifies to $a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a \leqslant 0$.
Thus, $(a+b+c)^{2} \leqslant 0$.
Clearly, $(a+b+c)^{2} \geqslant 0$.
Therefore, $(a+b+c)^{2}=0$, and hence, $a+b+c=0$.