JEE Main 2023 Mathematics question on Sequences and Series

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 $7^{\text{th}}$, $9^{\text{th}}$, $17^{\text{th}}$ 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 first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\dfrac{d}{12}$, is equal to __________.