1. (10 points) Simplify: $1 \div\left[a+1 \div\left(b+\frac{1}{\mathrm{c}}\right)\right] \div \frac{\mathrm{b}}{\mathrm{ab}+1}-\frac{1}{\mathrm{~b}(\mathrm{abc}+\mathrm{a}+\mathrm{c})}=$
【Analysis】This problem first converts division into multiplication, extracts common factors, and finally simplifies by cancellation.
【Solution】Solve: $1 \div\left[a+1 \div\left(b+\frac{1}{c}\right)\right] \div \frac{\mathrm{b}}{\mathrm{ab}+1}-\frac{1}{\mathrm{~b}(\mathrm{abc}+\mathrm{a}+\mathrm{c})}$
$$
\begin{array}{l}
=\frac{1}{a+\frac{c}{b c+1}} \div \frac{b}{a b+1}-\frac{1}{b(a b c+a+c)} \\
=\frac{b c+1}{a b c+a+c} \times \frac{a b+1}{b}-\frac{1}{b(a b c+a+c)} \\
=\frac{a b b^{2} c+b c+a b+1-1}{b(a b c+a+c)} \\
=\frac{b(a b c+a+c)}{b(a b c+a+c)} \\
=1
\end{array}
$$

Therefore, the answer is: 1.