Three, (20 points) (1) Given $a+\log _{2}(2 a+6)=11$ and $b+2^{b-1}=14$. Find the value of $a+b$.
(2) Given $f(x)=\frac{2}{2^{x-2}+1}$. Find
$$
f(-1)+f(0)+f(1)+f(2)+f(3)+f(4)+f(5)
$$

the value.
(1) Let $t=\log _{2}(2 a+6)$, then we have $t+2^{t-1}=14$. Since the function $f(x)=x+2^{x-1}$ is an increasing function on $\mathbf{R}$, we have $b$ $=t$. Therefore, $a+t=11$, which means $a+b=11$.
(2) Since $f(x)+f(4-x)=\frac{2}{2^{x-2}+1}+\frac{2}{2^{2-x}+1}$
$$
=\frac{2\left(2^{x-2}+1\right)}{2^{x-2}+1}=2 \text {, }
$$

then $f(-1)+f(0)+f(1)+f(2)+f(3)+f(4)+f(5)=7$.