CF1781H2.Window Signals (hard version)

This is the hard version of the problem. In this version, the constraints on $h$ and $w$ are higher.

A house at the sea has $h$ floors, all of the same height. The side of the house facing the sea has $w$ windows at equal distances from each other on every floor. Thus, the windows are positioned in cells of a rectangular grid of size $h \times w$.

In every window, the light can be turned either on or off, except for the given $k$ (at most $2$) windows. In these $k$ windows the light can not be turned on, because it is broken.

In the dark, we can send a signal to a ship at sea using a configuration of lights turned on and off. However, the ship can not see the position of the lights with respect to the house. Thus, if one configuration of windows with lights on can be transformed into another using parallel translation, these configurations are considered equal. Note that only parallel translation is allowed, but neither rotations nor flips are. Moreover, a configuration without any light at all is not considered valid.

Find how many different signals the ship can receive and print this number modulo $998\,244\,353$.

这是该问题的困难版本。在此版本中，$h$ 和 $w$ 的约束更大。

一座临海的房屋共有 $h$ 层，每层高度相同。房屋朝向大海的一侧，每层均有 $w$ 扇窗户，且各层窗户彼此间距相等。因此，所有窗户位于一个 $h \times w$ 大小的矩形网格的格点上。

每扇窗户中的灯均可开启或关闭，但给定的 $k$（最多为 $2$）扇窗户除外：这些窗户的灯无法开启（已损坏）。

在黑暗中，我们可通过控制各窗户灯的亮灭状态，向海上船只发送信号。然而，船只无法识别灯光相对于房屋的位置。因此，若一种亮灯配置可通过平移变换得到另一种亮灯配置，则这两种配置被视为等价。注意：仅允许平移，不允许旋转或翻转。此外，全都不亮的配置不被视为有效信号。

求船只可接收到的不同信号数量，并将结果对 $998\,244\,353$ 取模后输出。

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 100$). The description of the test cases follows.

The first line of each test case contains three integers $h$, $w$, and $k$ ($1 \le h, w \le 200$; $0 \le k \le \min(h \cdot w, 2)$) — the height of the house, the number of windows on each floor, and the number of windows with broken lights.

If $k \gt 0$, each of the next $k$ lines contains two integers $r_i$ and $c_i$ ($1 \le r_i \le h$; $1 \le c_i \le w$) — the floor number and the window number with the broken light. The floors are numbered $1$ to $h$ from bottom to top, and the windows are numbered $1$ to $w$ from left to right. If $k = 2$, then either $r_1 \ne r_2$ or $c_1 \ne c_2$.

It is guaranteed that the sum of $h \cdot w$ over all test cases does not exceed $40\,000$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1 \le t \le 100$）。随后是各测试用例的描述。

每个测试用例的第一行包含三个整数 $h$、$w$ 和 $k$（$1 \le h, w \le 200$；$0 \le k \le \min(h \cdot w, 2)$），分别表示房屋的高度、每层楼的窗户数量以及灯损坏的窗户数量。

若 $k > 0$，则接下来的 $k$ 行中，每行包含两个整数 $r_i$ 和 $c_i$（$1 \le r_i \le h$；$1 \le c_i \le w$），表示灯损坏的窗户所在的楼层号和窗户号。楼层编号从下到上为 $1$ 至 $h$，窗户编号从左到右为 $1$ 至 $w$。若 $k = 2$，则满足 $r_1 \ne r_2$ 或 $c_1 \ne c_2$。

保证所有测试用例的 $h \cdot w$ 之和不超过 $40\,000$。

For each test case, print a single integer — the number of signals the ship can receive, modulo $998\,244\,353$.

对于每个测试用例，输出一个整数——飞船能够接收到的信号数量对 $998\,244\,353$ 取模的结果。

In the first test case, the ship can receive four different signals: all lights on; lights on in two neighboring windows; lights on in the leftmost and the rightmost windows; or light on in any single window.

在第一个测试用例中，飞船可以接收到四种不同的信号：所有灯均亮起；两个相邻窗口的灯亮起；最左侧和最右侧窗口的灯亮起；或任意单个窗口的灯亮起。