For natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules. (가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively. (나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same. For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]
For natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules.\\
(가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively.\\
(나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same.\\
For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]