Mathematics Question from JEE Main 2023 · 25 January, Shift 1

Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A, A+1, A+2$, respectively. Let $a, b, c$ be the $7$th, $9$th, and $17$th terms of $A_1, A_2, A_3$, respectively, such that

$$\begin{vmatrix} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix} + 70 = 0$$

If $a = 29$, then the sum of the first $20$ terms of an AP whose first term is $c - a - b$ and common difference is $\frac{d}{12}$, is equal to _____.