<Question>
What is the distance, in units, between the points $(2, -6)$ and $(-4, 3)$? Express your answer in simplest radical form.
</Question>

<think>
<Plan_1>
Okay, so I need to find the distance between the two points (2, -6) and (-4, 3). Hmm, distance between two points in a plane. Let me recall how to do that. I think there's a formula for it, right?

<prompt_1.1> Recall and verify the distance formula for two points in a plane. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Maybe the distance formula? Yeah, something with the square root of the sum of the squares of the differences in coordinates. Let me try to remember. I think it's sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Wait, is that right? So you take the differences in the x-coordinates and the y-coordinates, square both of them, add them up, and then take the square root.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me check that.
If I have two points, say (x1, y1) and (x2, y2), then the distance between them is indeed sqrt[(x2 - x1)^2 + (y2 - y1)^2].

Okay, so let's apply that here. The points given are (2, -6) and (-4, 3). Let me label them.

<prompt_2.1> Apply the distance formula to the points (2, -6) and (-4, 3) and simplify the expression. </prompt_2.1>
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Wait, but let me visualize this on a coordinate plane to make sure I didn't make a mistake. 

If I plot the point (2, -6), that's in the fourth quadrant. Then (-4, 3) is in the second quadrant. The horizontal distance between them is from x=2 to x=-4.
So moving from 2 to -4, that's a distance of 2 - (-4) = 6 units to the left? Wait, no, actually, the difference in x is -4 - 2 = -6, so the absolute difference is 6. Similarly, the vertical distance from y=-6 to y=3 is 3 - (-6) = 9 units. So the horizontal distance is 6 and vertical distance is 9.
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Is there another way to check? Maybe using the Pythagorean theorem. If I consider the line between the two points as the hypotenuse of a right triangle, then the legs would be the horizontal and vertical distances. So, the legs are 6 and 9, and the hypotenuse should be sqrt(6^2 + 9^2) = sqrt(36 + 81) = sqrt(117). Which is the same as before.
-----

Wait, but just to make sure I didn't mix up the coordinates. The first point is (2, -6), so x1 is 2, y1 is -6. The second point is (-4, 3), so x2 is -4, y2 is 3.

<prompt_2.2> Reconfirm the application of the distance formula to the points (2, -6) and (-4, 3). </prompt_2.2>

Alternatively, maybe someone could confuse themselves by mixing up the order of subtraction, but as long as both differences are squared, the order doesn't matter because squaring removes any negative sign. Let me check that.

<prompt_2.3> Explore potential errors in the subtraction order and confirm the invariance of the squared differences. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's see. Let me call (2, -6) as (x1, y1) and (-4, 3) as (x2, y2). So plugging into the formula, the distance should be sqrt[(-4 - 2)^2 + (3 - (-6))^2]. Let me compute the differences first.

For the x-coordinates: x2 - x1 = -4 - 2 = -6.
For the y-coordinates: y2 - y1 = 3 - (-6) = 3 + 6 = 9. 

So then, squaring these differences: (-6)^2 = 36 and 9^2 = 81. Adding them together: 36 + 81 = 117. Then taking the square root of 117. So the distance is sqrt(117).
</execution_2.1>

<execution_2.2>
So when plugging into the formula, it's (x2 - x1) which is -4 - 2, yes, that's -6. And y2 - y1 is 3 - (-6) which is 9.
</execution_2.2>

<execution_2.3>
For example, (x2 - x1)^2 is the same as (x1 - x2)^2 because if you reverse the subtraction, you get a negative number, but squaring it makes it positive. So whether you do x2 - x1 or x1 - x2, the result is the same when squared. Similarly for y2 - y1 and y1 - y2. So the order of subtraction doesn't affect the final result.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
But the problem says to express the answer in simplest radical form. Hmm, sqrt(117) might not be simplified yet. Let me see if 117 has any perfect square factors.
Let's factor 117. 117 divided by 9 is 13. So 117 = 9 * 13. Since 9 is a perfect square, sqrt(117) = sqrt(9*13) = sqrt(9)*sqrt(13) = 3*sqrt(13).
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Wait a second, but just to ensure that I didn't make a mistake in the subtraction steps. Let's recalculate the differences.

<prompt_3.1> Recalculate the differences in coordinates and verify the simplification of the square root. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
x2 - x1: -4 - 2 = -6. Correct. Then squared is 36. 

y2 - y1: 3 - (-6) = 3 + 6 = 9. Correct. Squared is 81. 

36 + 81 = 117. Correct. Then sqrt(117) simplifies to 3*sqrt(13). Right, because 117 is 9*13. 9 is the largest perfect square factor here.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
So the answer should be 3√13 units. Let me just verify once more. If I have a right triangle with legs 6 and 9, hypotenuse squared is 6² + 9² = 36 + 81 = 117, so hypotenuse is √117. Simplify that, √(9*13) = 3√13. Yep, that's correct.
-----

I think that's thorough.
I checked multiple steps, verified the formula, recalculated the differences, confirmed the simplification of the radical. All steps lead to the same conclusion.
</Plan_4>

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

<Final_answer>
Therefore, the distance between the points $(2, -6)$ and $(-4, 3)$ is $\boxed{3\sqrt{13}}$ units.
</Final_answer>
</think>