<Question>
The point $(0,0)$ is reflected over the vertical line $x=1$.  When its image is then reflected over the line $y=2$, what is the resulting point?

Write your answer in the form $(x, y)$ where $x$ and $y$ are real numbers.
</Question>

<think>
<Plan_1>
Okay, so I need to find the resulting point after reflecting (0,0) first over the vertical line x=1 and then reflecting that image over the horizontal line y=2. Hmm, let me think step by step.

First, reflection over a vertical line. I remember that reflecting a point over a vertical line x=a changes the x-coordinate based on the distance from the line. The formula, if I recall correctly, is something like (2a - x, y). Let me verify that.
If you have a point (x, y) and you reflect it over x=a, the distance from the point to the line is |x - a|, so the image should be on the opposite side of the line at the same distance. So the new x-coordinate would be a + (a - x) = 2a - x. The y-coordinate remains the same. So yes, the formula is (2a - x, y).

So applying that to the first reflection over x=1.

<prompt_1.1> Use the reflection formula over the vertical line x=1 to find the image of the point (0,0). </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The original point is (0,0). Here, a=1, x=0, y=0. So the new x-coordinate is 2*1 - 0 = 2. The y-coordinate remains 0. So the image after the first reflection is (2,0).
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, we need to reflect this point (2,0) over the horizontal line y=2. For reflections over horizontal lines, the formula should be similar but for the y-coordinate. If reflecting over y=b, the formula is (x, 2b - y). Let me check.
Again, the distance from the point to the line is |y - b|, so the image should be the same distance on the other side. So new y-coordinate is b + (b - y) = 2b - y. The x-coordinate remains the same.

<prompt_2.1> Use the reflection formula over the horizontal line y=2 to find the image of the point (2,0). </prompt_2.1>
-----

Wait, let me visualize this to make sure.

<prompt_2.2> Visualize the reflections to confirm the resulting point. </prompt_2.2>

Alternatively, maybe I can think of another way. If I first reflect over x=1, which moves (0,0) to (2,0), then reflecting over y=2.
So vertical distance from (2,0) to y=2 is 2 units. So moving 2 units above y=2 would be y=4. So the point would be (2,4).

Alternatively, maybe using coordinate transformations. The reflection over x=1 can be thought of as translating the coordinate system so that x=1 becomes the origin, then reflecting, then translating back. But maybe that's complicating it.
Let me try.

<prompt_2.3> Explore alternative methods to confirm the resulting point. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Applying that to our case, the point is (2,0) and the line is y=2. So the new y-coordinate is 2*2 - 0 = 4. The x-coordinate remains 2. So the resulting point after the second reflection is (2,4).
</execution_2.1>

<execution_2.2>
Starting at (0,0), reflecting over x=1. The line x=1 is vertical, so reflection over it would flip the x-coordinate across that line. So since 0 is 1 unit away from 1 (to the left), the reflection would be 1 unit to the right of 1, which is 2. So (2,0). Then reflecting over y=2. The line y=2 is horizontal. The point (2,0) is 2 units below y=2, so reflecting it would put it 2 units above, which is y=4. So yes, (2,4).
</execution_2.2>

<execution_2.3>
For reflection over x=1:

1. Translate the original coordinate system so that x=1 is at the origin. This is done by subtracting 1 from the x-coordinate. So (0,0) becomes (-1, 0).
2. Reflect over the new origin (which was x=1 in the original). Reflection over the y-axis (which is the new origin) would change (x, y) to (-x, y). So (-1, 0) becomes (1, 0).
3. Translate back by adding 1 to the x-coordinate: (1 + 1, 0) = (2, 0). That matches the previous result.

Similarly, reflecting over y=2:

1. Translate the coordinate system so that y=2 is the origin. Subtract 2 from the y-coordinate. So (2,0) becomes (2, -2).
2. Reflect over the new origin (which was y=2). Reflection over the x-axis (since y=2 is now the origin, reflecting over it would invert the y-coordinate). So (2, -2) becomes (2, 2).
3. Translate back by adding 2 to the y-coordinate: (2, 2 + 2) = (2, 4). Wait, no. Wait, translating back would mean adding 2 to the y-coordinate. But the reflection over y=2 in the translated system would have been (2, -2) reflecting to (2, 2), then translating back to (2, 2 + 2) = (2,4). Hmm, that's another way of looking at it, but actually, if you translate the point (2, -2) back by adding 2 to y, it becomes (2, 0), which is not correct. Wait, maybe my steps here are off.

Wait, maybe the translation steps need to be carefully followed. Let me clarify:

When translating the coordinate system so that y=2 becomes the origin, you subtract 2 from the y-coordinate. So any point (x, y) in the original system becomes (x, y - 2) in the translated system. Then reflecting over the new origin (which is y=2 in the original) would reflect (x, y - 2) to (x, -(y - 2)) = (x, -y + 2). Then translating back by adding 2 to the y-coordinate: (x, (-y + 2) + 2) = (x, -y + 4). Wait, so the reflection over y=2 would be (x, 4 - y). So for the point (2,0), reflecting over y=2 would give (2, 4 - 0) = (2,4). That's correct. So that's another way to see it. So the formula is (x, 4 - y). So yes, (2,0) becomes (2,4).

So that seems correct.

Alternatively, if you use the formula for reflection over y=2, which is (x, 2*2 - y) = (x, 4 - y). So 4 - 0 = 4. So (2,4). Correct.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, is that correct? Let me check again.

<prompt_3.1> Recheck the reflection over the horizontal line y=2 and confirm the final result. </prompt_3.1>
-----

Based on execution_2.3:
Hmm, so I think that's correct.

<prompt_3.2> Finalize the solution and confirm the resulting point. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
First reflection: (0,0) over x=1. The horizontal distance from x=0 to x=1 is 1 unit. Reflecting over x=1 would mirror that distance on the other side, so from x=1 to x=2. So yes, (2,0). Then reflecting over y=2. The vertical distance from y=0 to y=2 is 2 units. Reflecting over y=2 would mean moving 2 units above y=2, which is y=4. So the point (2,0) becomes (2,4).
</execution_3.1>

<execution_3.2>
But let me check if there's another way to see this. Maybe using vectors or geometry.

Alternatively, the composition of two reflections. But maybe that's overcomplicating. Since each reflection is straightforward, step by step, the final result is (2,4). I don't see any errors in my reasoning. So I think that's the correct answer.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
**Final Answer**
The resulting point is \boxed{(2, 4)}.
</Plan_4>

<Execution_4>
No prompt to execute.
</Execution_4>

<Final_answer>
Therefore, the resulting point after the two reflections is $\boxed{(2,4)}$.
</Final_answer>
</think>