Matrices and Determinants: JEE Main 2025 Mathematics question

Let the system of equations $x + 5y - z = 1$, $4x + 3y - 3z = 7$, and $24x + y + \lambda z = \mu$, with $\lambda,\mu \in \mathbb{R}$, have infinitely many solutions. Then the number of solutions of this system, if $x,y,z$ are integers and satisfy $7 \le x+y+z \le 77$, is: