P11168 "CMOI R1" First Town of This Journey / Grid Covering.

$\small\color{white}/10^{\text{th}}\text{Problem by AtC}.$ In this problem, the line connecting two points is considered to be the line segment with these two points as endpoints.

$\text{BiOI}$ has a grid with $n$ rows and $m$ columns. You need to choose as few lattice points as possible so that, for any two **distinct** lattice points, the line segment connecting them passes through at least one chosen point (here we consider that a segment **passes through** its two endpoints). $\text{CmOI}$ thinks this problem is too easy, so he additionally gives $a, b, x$, meaning that the lattice point at row $a$, column $b$ must be chosen or must not be chosen: * When $x = 0$, this lattice point cannot be chosen. * When $x = 1$, this lattice point must be chosen. You only need to output the minimum number of lattice points to choose. $\text{BiOI}$ and $\text{CmOI}$ will give the grid and your answer to $\text{ArCu}$ so that he can construct an actual selection.

The first line contains two integers $n, m$. The second line contains three integers $a, b, x$.

Output one integer in one line, the minimum number of lattice points that need to be chosen.

$\text{Details about Subtasks}:$ All testdata satisfy $1 \le n, m < 2^{32}$, $1 \le a \le n$, $1 \le b \le m$, $0 \le x \le 1$. * $\text{Subtask 1}: 1 \le n, m \le 10, \text{5 points.}$ * $\text{Subtask 2}: x = 0, \text{25 points}.$ * $\text{Subtask 3}: x = 1, \text{30 points.}$ * $\text{Subtask 4}: \text{40 points.}$ $\text{Sample Explanation}:$ * $\text{Sample 1}:$ The grid has $3$ rows and $3$ columns, and the lattice point at row $2$, column $2$ must be chosen. An optimal solution is as follows (black points are chosen lattice points):  Note that the following solutions are not valid: * The figure shows two ways where the segment between two distinct lattice points does not pass through any black point. * The lattice point at row $2$, column $2$ is not chosen, but the input requires this point to be chosen.  Also, we consider that having an endpoint on a black point counts as passing through a black point. That is, the following segment passes through a black lattice point.  Translated by ChatGPT 5