As shown in the figure, a square A with side length 2 and a square B with side length 1 have sides parallel to each other, and the intersection point of the two diagonals of A coincides with the intersection point of the two diagonals of B. Let R be the region of A and its interior excluding the interior of B. For a natural number $n \geqq 2$, small squares with side length $\frac { 1 } { n }$ are drawn in R according to the following rule. (가) One side of each small square is parallel to a side of A. (나) The interiors of the small squares do not overlap with each other. According to such rules, let $a _ { n }$ be the maximum number of small squares with side length $\frac { 1 } { n }$ that can be drawn in R. For example, $a _ { 2 } = 12$ and $a _ { 3 } = 20$. When $\lim _ { n \rightarrow \infty } \frac { a _ { 2 n + 1 } - a _ { 2 n } } { a _ { 2 n } - a _ { 2 n - 1 } } = c$, find the value of $100 c$. [4 points]
As shown in the figure, a square A with side length 2 and a square B with side length 1 have sides parallel to each other, and the intersection point of the two diagonals of A coincides with the intersection point of the two diagonals of B. Let R be the region of A and its interior excluding the interior of B.
For a natural number $n \geqq 2$, small squares with side length $\frac { 1 } { n }$ are drawn in R according to the following rule.\\
(가) One side of each small square is parallel to a side of A.\\
(나) The interiors of the small squares do not overlap with each other.
According to such rules, let $a _ { n }$ be the maximum number of small squares with side length $\frac { 1 } { n }$ that can be drawn in R. For example, $a _ { 2 } = 12$ and $a _ { 3 } = 20$. When $\lim _ { n \rightarrow \infty } \frac { a _ { 2 n + 1 } - a _ { 2 n } } { a _ { 2 n } - a _ { 2 n - 1 } } = c$, find the value of $100 c$. [4 points]