As shown in the figure below, from a square with side length 1, a square with side length $\frac { 1 } { 2 }$ is cut out, and the remaining shape is called $A _ { 1 }$. From a square with side length $\frac { 1 } { 4 }$, a square with side length $\frac { 1 } { 8 }$ is cut out, and two resulting shapes are attached to the upper two sides of $A _ { 1 }$ to form a figure called $A _ { 2 }$. From a square with side length $\frac { 1 } { 16 }$, a square with side length $\frac { 1 } { 32 }$ is cut out, and four resulting shapes are attached to the upper four sides of $A _ { 2 }$ to form a figure called $A _ { 3 }$. Continuing this process, let $A _ { n }$ be the $n$-th figure obtained and $S _ { n }$ be its area. If $\lim _ { n \rightarrow \infty } S _ { n } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
As shown in the figure below, from a square with side length 1, a square with side length $\frac { 1 } { 2 }$ is cut out, and the remaining shape is called $A _ { 1 }$.\\
From a square with side length $\frac { 1 } { 4 }$, a square with side length $\frac { 1 } { 8 }$ is cut out, and two resulting shapes are attached to the upper two sides of $A _ { 1 }$ to form a figure called $A _ { 2 }$.\\
From a square with side length $\frac { 1 } { 16 }$, a square with side length $\frac { 1 } { 32 }$ is cut out, and four resulting shapes are attached to the upper four sides of $A _ { 2 }$ to form a figure called $A _ { 3 }$.
Continuing this process, let $A _ { n }$ be the $n$-th figure obtained and $S _ { n }$ be its area.\\
If $\lim _ { n \rightarrow \infty } S _ { n } = \frac { q } { p }$, find the value of $p + q$.\\
(Here, $p$ and $q$ are coprime natural numbers.) [4 points]