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\section{Models}  \subsection{Shot generation}  Each shot was modeled as a circle. The center of each circle was positioned using a random walk. A random walk is a mathematical formalization of a path that consists of a succession of random steps. This model is constrained For a system of $k$ circles, $k$ random walks were performed  in that separate circles cannot coincide. the Python programming language. The radius from the center cell in the lattice was then determined for each circle.  Thus, in Let $r$ denote the radius of each circle (corresponding either to $2$, $4$, $7$, or $9$ cells); let $s$ denote the distance from the center of one circle to the center of another.  In  this application, a random walk of length $N$ in a 100✕100 $2$-dimensional integer lattice ${\Bbb Z}^2$ (also known as a square lattice), starting at point $x$, is defined as a path \begin{equation}  ω=(ω_0, ω_1, ..., ω_n)  \end{equation}  In this case, $ω_b ∈ {\Bbb Z}^2$, $ω_0=x$, $|ω_b−ω_{b−1}| = 1$, where  $b = 1, 2, . . . , n$,$ω_a \neq ω_b$ for $a \neq b$,  and $0 ≤ a < b ≤ n$. $r_0+r_1 \neq s$.  Let $|ω| = N$ denote the length of $ω$. To set these constraints we took into account the radi of the circles and center of the circles. The constraint states that the distance between the centers of the circles, as defined by the distance equation, cannot be less than the sum of the radii of these respective circles. To implement this into an equation, given $r$ is the radius of circle $z$ and $(x,y)$ is the center of the circle on the premade 100x100 grid, the constraint is equal to: