For a natural number $n$, point $\mathrm { A } _ { n }$ is a point on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$. (2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting point $\mathrm { P } _ { n }$ about the line $y = x$. (3) Let $\mathrm { R } _ { n }$ be the point where the line passing through point $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis. (4) Let $\mathrm { A } _ { n + 1 }$ be the point obtained by translating point $\mathrm { R } _ { n }$ by 1 unit in the direction of the $x$-axis. Let the $x$-coordinate of point $\mathrm { A } _ { n }$ be $x _ { n }$. When $x _ { 5 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]
For a natural number $n$, point $\mathrm { A } _ { n }$ is a point on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule.\\
(가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$.\\
(나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.\\
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting point $\mathrm { P } _ { n }$ about the line $y = x$.\\
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through point $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.\\
(4) Let $\mathrm { A } _ { n + 1 }$ be the point obtained by translating point $\mathrm { R } _ { n }$ by 1 unit in the direction of the $x$-axis.\\
Let the $x$-coordinate of point $\mathrm { A } _ { n }$ be $x _ { n }$. When $x _ { 5 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]