There are N rectangles numbered from $0$ to N$-1.$ The K$-$th rectangle has size A$[$K$]$ x B$[$K$]$

We call two rectangles similar when:

they share a side of the same length, and

the length of the other sides differs by at most $1.$

For example:

rectangles $2$x$3$ and $2$x$4$ are similar,

rectangles $2$x$3$ and $4$x$2$ are also similar $($rectangles can be rotated$),$

rectangles $2$x$3$ and $2$x$5$ are not similar $($other sides differ by $2).$

rectangles $2$x$2$ and $3$x$3$ are not similar either $($more than one side differing in length$).$

The task is to choose as many rectangles as possible in such a way that every two chosen rectangles are similar to each other. Note that rectangles can be rotated.

Write a function:

class Solution $\{$ public int solution$($int$[]$ A$,$ int$[]$ B$)$; $\}$

that, given two arrays A and B of N integers each, returns the maximum number of similar rectangles that can be chosen.

Examples:

A $1.$ Given A $(500,500)$ and B $(500,500),$ the function should return $2.$ Both rectangles can be chosen.

$2.$ Given A$(5,5,1,6,4)$ and B $(2,1\mathrm{.}6,1,1]$ the function should return $3.$ The best choice is to take the rectangles in positions $1$ to $3.$ After rotation, their sizes are respectively $5$x$1,61$ and $6$x$1.$

$3.$ Given A $(3,3,3,3,5,7,1,1)$ and B $(4,6,8,10,10,10,2,3)$ the function should return $2.$ The optimal choice is to take last two rectangles $($sizes $1$x$2$ and $1$x$3)$

$4.$ Given A $(4,4,2,2,2,4,4)$ and B $=13,4,6,6,6,3,4]$ the function should return $4.$ The best choice is to take the first two and the last two rectangles $($sizes $3$x$4,4$x$4,3$x$4$ and $4$x$4).$

Write an efficient algorithm for the following assumptions

N is an integer within the range $[1.\mathrm{.}200,000]$;

arrays A and B consist of N integers within the range $[1.\mathrm{.}500]$