D. Counting Pointstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThe pink soldiers drew $$$n$$$ circles with their center on the $$$x$$$-axis of the plane. Also, they have told that the sum of radii is exactly $$$m$$$$$$^{\text{∗}}$$$.Please find the number of integer points inside or on the border of at least one circle. Formally, the problem is defined as follows.You are given an integer sequence $$$x_1,x_2,\ldots,x_n$$$ and a positive integer sequence $$$r_1,r_2,\ldots,r_n$$$, where it is known that $$$\sum_{i=1}^n r_i = m$$$.You must count the number of integer pairs $$$(x,y)$$$ that satisfy the following condition. There exists an index $$$i$$$ such that $$$(x-x_i)^2 + y^2 \le r_i^2$$$ ($$$1 \le i \le n$$$). $$$^{\text{∗}}$$$Is this information really useful? Don't ask me; I don't really know.InputEach test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows. The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n \le m \le 2\cdot 10^5$$$).The second line of each test case contains $$$x_1,x_2,\ldots,x_n$$$ — the centers of the circles ($$$-10^9 \le x_i \le 10^9$$$).The third line of each test case contains $$$r_1,r_2,\ldots,r_n$$$ — the radii of the circles ($$$1 \le r_i$$$, $$$\sum_{i=1}^n r_i = m$$$).It is guaranteed that the sum of $$$m$$$ over all test cases does not exceed $$$2\cdot 10^5$$$.OutputFor each test case, output the number of integer points satisfying the condition on a separate line.ExampleInput42 30 01 22 30 21 23 30 2 51 1 14 80 5 10 152 2 2 2Output13
16
14
52
NoteOn the first test case, the circle with $$$r_1=1$$$ is completely inside the circle with $$$r_2=2$$$. Therefore, you only have to count the number of integer points inside the latter. There are $$$13$$$ integer points such that $$$x^2+y^2 \le 2^2$$$, so the answer is $$$13$$$.On the second test case, the circle with $$$r_1=1$$$ is not completely inside the circle with $$$r_2=2$$$. There are $$$3$$$ additional points that are inside the first circle but not inside the second circle, so the answer is $$$3+13=16$$$.