Let $a, b, c$ be real numbers such that the following system of equations has a solution
\[\begin{aligned}
x+2 y+3 z &=a & & (1)\\
8 x+10 y+12 z &=b & & (2)\\
7 x+8 y+9 z &=c-1 & & (3)
\end{aligned}\]
Let $A$ be a matrix such that
\[A=\left[\begin{array}{ccc}
a & 1 & c \\
b & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]\]
What is the value of $\operatorname{det}(A)$?
- $1$
- $2$
- $3$
- $4$
- $5$