<Question>
Consider the rectangular region with the following points as vertices: $$(5,4), (-5,4), (-5,-4), (5,-4).$$ How many points with integer coordinates will be strictly in the interior of this rectangular region?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the number of integer coordinate points that are strictly inside a rectangle with vertices at (5,4), (-5,4), (-5,-4), and (5,-4). Hmm, first, I should probably visualize the rectangle to understand its dimensions and position on the coordinate plane.

<prompt_1.1> Identify the range of x and y coordinates for the interior points of the rectangle based on the given vertices. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, the vertices are given as (5,4), (-5,4), (-5,-4), and (5,-4). Let me plot these points mentally. The top side of the rectangle is from (-5,4) to (5,4), so that's a horizontal line 10 units long, right? Because the distance between -5 and 5 on the x-axis is 10. Similarly, the bottom side is from (-5,-4) to (5,-4), which is also 10 units. The vertical sides are from y=4 to y=-4, which is a total vertical distance of 8 units. So the rectangle is 10 units wide (along the x-axis) and 8 units tall (along the y-axis).

Now, the problem asks for the number of integer coordinate points strictly inside this rectangle. "Strictly inside" means that the points cannot be on the edges or vertices of the rectangle. So, they have to be inside the boundaries.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me recall that for a rectangle with integer coordinates, the number of interior points can be found by considering the grid points that lie strictly between the left, right, top, and bottom boundaries.

<prompt_2.1> Calculate the number of interior points for the x and y coordinates based on the range identified. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
First, let's determine the range for x and y coordinates of the interior points.

The rectangle extends from x = -5 to x = 5. But since the points on the edges are excluded, the interior x-coordinates must be greater than -5 and less than 5. Similarly, the y-coordinates range from y = -4 to y = 4, but excluding the edges, so they must be greater than -4 and less than 4.

Wait, but hold on. The vertices are at (5,4), etc. , so the rectangle includes all points from x = -5 to x = 5 and y = -4 to y = 4. But strictly inside would mean x must be between -5 and 5, but not equal to -5 or 5, and y must be between -4 and 4, but not equal to -4 or 4.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
So, for the x-coordinates: the integer values strictly between -5 and 5. Let's list them.

<prompt_3.1> List the integer x-coordinates strictly between -5 and 5. </prompt_3.1>

Similarly, for the y-coordinates: integer values strictly between -4 and 4.

<prompt_3.2> List the integer y-coordinates strictly between -4 and 4. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting from -4, -3, -2, -1, 0, 1, 2, 3, 4. Wait, because -5 is excluded, the next integer is -4, and 5 is excluded, so the last integer is 4. So that's from -4 to 4 inclusive. How many integers is that? Let's count: from -4 to 4, that's 9 numbers (-4, -3, -2, -1, 0, 1, 2, 3, 4).
</execution_3.1>

<execution_3.2>
Let's list them.

Starting with x-coordinates: greater than -5 and less than 5. The integers in this range are -4, -3, -2, -1, 0, 1, 2, 3, 4. Let me count them: from -4 up to 4, that's 9 numbers.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_3.2:
Similarly, for y-coordinates: strictly between -4 and 4.
So integers greater than -4 and less than 4. That would be -3, -2, -1, 0, 1, 2, 3. Wait, let's check:

<prompt_4.1> Calculate the total number of interior points by multiplying the number of x and y coordinates. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
If y must be greater than -4 and less than 4, then starting at -3 (since -4 is excluded), up to 3 (since 4 is excluded). So the integers are -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

Therefore, the number of interior points would be the product of the number of possible x-coordinates and y-coordinates. So 9 (x) multiplied by 7 (y) gives 9*7=63 points.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Wait, but let me verify this. Because sometimes when dealing with coordinates, it's easy to make a mistake in counting. Let me double-check the x and y counts.

<prompt_5.1> Verify the calculation of the number of interior points by re-evaluating the range and count of x and y coordinates. </prompt_5.1>

But hold on, sometimes when you have a rectangle aligned with the grid, the number of interior points can also be calculated using the formula: (width - 2) * (height - 2), where width and height are the number of units in each direction. Wait, but let me confirm.

<prompt_5.2> Use an alternative method to verify the number of interior points using the formula (width - 2) * (height - 2). </prompt_5.2>

Alternatively, sometimes people use the floor or ceiling functions, but in this case, since the coordinates are integers, it's straightforward.

But let me make sure that the original rectangle is indeed axis-aligned.

<prompt_5.3> Confirm that the rectangle is axis-aligned and verify the calculation of interior points. </prompt_5.3>

Alternatively, another way to check: let's consider a smaller rectangle. Suppose we have a rectangle from (-1, -1) to (1,1).

<prompt_5.4> Verify the formula for interior points using smaller rectangles as examples. </prompt_5.4>
</Plan_5>

<Execution_5>
<execution_5.1>
For the x-direction: the rectangle spans from x = -5 to x = 5. Excluding the endpoints, the interior x-values are from -4 to 4. How many integers are there between -4 and 4 inclusive? Let's calculate: from -4 to 4, the number of integers is 4 - (-4) + 1 = 8 + 1 = 9. Yes, that's correct. So 9 x-coordinates.

For the y-direction: the rectangle spans from y = -4 to y = 4. Excluding the endpoints, the interior y-values are from -3 to 3. Let's check: -3 to 3 inclusive. The number of integers is 3 - (-3) + 1 = 6 + 1 = 7. Correct, 7 y-coordinates.

Therefore, the total number of interior points is indeed 9 * 7 = 63.
</execution_5.1>

<execution_5.2>
The width in the x-direction is from -5 to 5, which is a total of 11 units (5 - (-5) = 10, but the number of integer points would be 11: from -5 to 5 inclusive). But since we are excluding the endpoints, the number of interior x-coordinates is 10 - 2 = 9? Wait, 5 - (-5) is 10 units, but the number of integer points is 11 (including both endpoints). Excluding the endpoints, we have 9 points, which matches our previous count.

Similarly, the height is from -4 to 4, which is 8 units. The number of integer points including endpoints is 9 (from -4 to 4). Excluding the endpoints, we have 7 points. So the formula would be (width in units - 2) * (height in units - 2). Wait, but here the width in units is 10 (from -5 to 5) and height is 8 (from -4 to 4). So (10 - 2)*(8 - 2) = 8*6=48. Wait, that's conflicting with our previous result. Hmm, so which is correct?

Wait, now I'm confused. There seems to be a discrepancy here. Let me clarify.
</execution_5.2>

<execution_5.3>
Looking at the vertices: (5,4), (-5,4), (-5,-4), (5,-4). Since the x and y coordinates are aligned with the axes, yes, the rectangle is axis-aligned. So its sides are parallel to the x and y axes.

Therefore, the number of interior points should be (right - left - 2) * (top - bottom - 2). Wait, let's see. The left x-coordinate is -5, right is 5. So the width is 5 - (-5) = 10 units. But the number of integer x-coordinates between -5 and 5, exclusive, is 9 (from -4 to 4). Similarly, the top y-coordinate is 4, bottom is -4. The height is 4 - (-4) = 8 units. The number of integer y-coordinates between -4 and 4, exclusive, is 7 (from -3 to 3). So 9*7=63.
</execution_5.3>

<execution_5.4>
Then the interior points would be (0,0), so 1 point. Using the formula (2 - 2)*(2 - 2) = 0*0=0, which is wrong. Wait, so that formula isn't correct. Hmm. Maybe the formula is (width - 2)*(height - 2) but in this case, the width is 2 units (from -1 to 1), so 2 units, so 2 - 2 = 0. That gives 0, which is wrong. So that formula isn't applicable here.

Alternatively, perhaps the formula is (number of interior x-coordinates) * (number of interior y-coordinates). So in this case, for the small square from -1 to 1 in both x and y, the interior x-coordinates would be 0 (only one), same for y. So 1*1=1, which is correct. So maybe the general formula is (number of interior x points) * (number of interior y points).

So in our original problem, the number of interior x points is 9 (from -4 to 4 inclusive), and the number of interior y points is 7 (from -3 to 3 inclusive). So 9*7=63.
</execution_5.4>
</Execution_5>

<Plan_6>
Based on execution_5.1, execution_5.2, execution_5.3, execution_5.4:
Therefore, the answer should be 63. But let me just confirm again by considering the coordinates.

<prompt_6.1> Perform a final verification of the calculation and conclude the number of interior points. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
For example, take x from -4 to 4 (9 values) and y from -3 to 3 (7 values). Each combination (x,y) is a unique interior point. So 9*7=63. That seems correct.

Alternatively, maybe I can visualize the grid. If I draw the rectangle from x=-5 to 5 and y=-4 to 4, the interior would exclude the borders. So the interior is from x=-4.5 to 4.5 (if the grid is integer points), but since we are considering integer coordinates, the x must be from -4 to 4 inclusive, and y from -3 to 3 inclusive. So yes, 9 x 7 = 63.

Therefore, after all these considerations and checks, I believe the correct answer is 63.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
**Final Answer**
The number of integer coordinate points strictly in the interior of the rectangular region is \boxed{63}.
</Plan_7>

<Execution_7>
No prompt to execute.
</Execution_7>

<Final_answer>
Thus, the number of points with integer coordinates that will be strictly in the interior of the rectangular region is $\boxed{63}$.
</Final_answer>
</think>