<Question>
What is the value of $513^2 - 487^2$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is to find the value of 513 squared minus 487 squared. Hmm, that seems like a difference of squares. Wait, yes, the formula for difference of squares is a² - b² = (a - b)(a + b). Maybe I can apply that here instead of calculating each square separately, which might be tedious.

<prompt_1.1> Apply the difference of squares formula to the expression 513² - 487². </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me check if that's right. So, if a is 513 and b is 487, then according to the formula, 513² - 487² should equal (513 - 487)(513 + 487). Let me compute those terms step by step.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
First, calculate the difference between 513 and 487.
Let me subtract 487 from 513. 513 minus 487... 500 minus 400 is 100, and 13 minus 87 is -74, so 100 - 74 is 26.
Wait, no, that's not right. Wait, 513 - 487. Let's do it properly.

<prompt_2.1> Calculate 513 - 487 to verify the previous result. </prompt_2.1>

Then, the sum of 513 and 487. Let's add those two numbers.
513 + 487. 500 + 400 is 900, and 13 + 87 is 100. So total is 900 + 100 = 1000. So the sum is 1000.

So, according to the difference of squares formula, the original expression simplifies to 26 multiplied by 1000.
So, 26 * 1000 = 26,000. Wait, is that correct? Let me verify.

<prompt_2.2> Verify the result by directly calculating 513² - 487². </prompt_2.2>
-----

Hmm, but let me just make sure I didn't make any arithmetic errors. Let me recheck the subtraction and addition.

<prompt_2.3> Recheck the subtraction 513 - 487 and addition 513 + 487. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Starting from 513 minus 487. 487 plus 26 is 513 because 487 + 20 is 507, then +6 is 513. So yes, 513 - 487 is 26.
</execution_2.1>

<execution_2.2>
Let me check by calculating 513 squared and 487 squared separately and then subtracting to see if it matches 26,000. 

Calculating 513 squared. Hmm, 500 squared is 250,000. Then, 513 is 13 more than 500. So, (500 + 13)² = 500² + 2*500*13 + 13². That's 250,000 + 13,000 + 169 = 250,000 + 13,000 is 263,000, plus 169 is 263,169. So 513 squared is 263,169.

Now, 487 squared. Let's see, 500 - 13 is 487. So (500 - 13)² = 500² - 2*500*13 + 13². That's 250,000 - 13,000 + 169. 250,000 - 13,000 is 237,000, plus 169 is 237,169. So 487 squared is 237,169.

Subtracting the two: 263,169 - 237,169. Let's do that. 263,169 minus 237,169. Subtract the hundreds of thousands first: 263,000 - 237,000 = 26,000. Then the remaining 169 - 169 = 0. So total is 26,000.
</execution_2.2>

<execution_2.3>
513 - 487: 487 + 26 = 513. Yes, because 487 + 20 is 507, then +3 is 510, then +3 more is 513. Wait, no. 487 + 26: 487 + 20 is 507, then +6 is 513. So yes, 26. That's correct. And 513 + 487: 500 + 400 is 900, 13 + 87 is 100, so 900 + 100 is 1000.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Alternatively, using the formula, (513 - 487)(513 + 487) = 26*1000 = 26,000. So both methods give the same result.
Therefore, the answer must be 26,000.

Based on execution_2.2, execution_2.3:
I think that's solid. No mistakes detected in the reasoning. So the value is 26,000.
</Plan_3>

<Execution_3>
No prompt to execute.
</Execution_3>

<Final_answer>
Therefore, the value of $513^2 - 487^2$ is $\boxed{26000}$.
</Final_answer>
</think>