<Question>
What is the sum of the values of $a$ that satisfy the equation $$(3)5^2-4(5-a)^2 \div 3=63?$$
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the sum of all values of 'a' that satisfy the equation (3)5² - 4(5 - a)² ÷ 3 = 63. Hmm, first, I should probably simplify the equation step by step. Let me write it down again to make sure I have it right: 3 times 5 squared minus 4 times (5 - a) squared divided by 3 equals 63. So, in mathematical terms, that's 3*(5²) - [4*(5 - a)²]/3 = 63.

Alright, let's start solving this step by step. Maybe I should handle the multiplication and division before dealing with the addition and subtraction. Let's compute 3*(5²) first.

<prompt_1.1> Simplify the equation by calculating 3*(5²). </prompt_1.1>
-----

But wait, let me check if there's another way this equation could be interpreted. For example, could the division by 3 apply only to the 4(5 - a)^2 term, which it does, and not to the entire 3*5²? The equation is written as (3)5² - 4(5 - a)^2 ÷ 3 = 63.

<prompt_1.2> Consider different interpretations of the equation and confirm the correct order of operations. </prompt_1.2>

Alternatively, maybe someone could misplace the division, but according to the original equation, the division by 3 is only on the 4(5 - a)^2 term. So, the equation is correctly interpreted as 3*25 - [4*(5 - a)^2]/3 = 63.
</Plan_1>

<Execution_1>
<execution_1.1>
5 squared is 25, so 3*25 is 75. So the equation simplifies to 75 - [4*(5 - a)²]/3 = 63.
</execution_1.1>

<execution_1.2>
Depending on how it's parenthesized. If it's 3*5² - (4*(5 - a)^2)/3 = 63, which is how I interpreted it. Since exponentiation and multiplication come before division in order of operations, but the division is specifically after the multiplication. So yes, I think my steps are correct.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to isolate the term with 'a'. Let's subtract 75 from both sides to get rid of that constant term on the left. So, subtracting 75 from both sides gives: -[4*(5 - a)²]/3 = 63 - 75. Calculating the right side: 63 - 75 is -12. So now the equation is -[4*(5 - a)²]/3 = -12.

Now, I can multiply both sides by -1 to make the equation look a bit nicer.
That would give [4*(5 - a)²]/3 = 12.

Next, to eliminate the fraction, I can multiply both sides by 3.

<prompt_2.1> Simplify the equation further and expand the squared term. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Multiplying both sides by 3 gives 4*(5 - a)² = 36. Then, divide both sides by 4 to simplify further. That would give (5 - a)² = 9.

So now, we have a squared term. To solve for 'a', we can take the square root of both sides. Remember, taking the square root will give both positive and negative solutions. So, sqrt[(5 - a)²] = sqrt[9], which simplifies to |5 - a| = 3. This absolute value equation means that 5 - a can be equal to 3 or 5 - a can be equal to -3.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let's solve both equations.

First, 5 - a = 3.

<prompt_3.1> Solve the equations 5 - a = 3 and 5 - a = -3 for 'a'. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Subtracting 5 from both sides gives -a = -2. Multiplying both sides by -1 gives a = 2.

Second, 5 - a = -3. Subtracting 5 from both sides gives -a = -8. Multiplying by -1 gives a = 8.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
So the solutions are a = 2 and a = 8. The problem asks for the sum of the values of 'a', so 2 + 8 = 10.
Therefore, the answer should be 10.

But let me verify my steps again to make sure I didn't make any mistakes.

Starting from the beginning: 3*5² - [4*(5 - a)²]/3 = 63. Calculated 3*25 = 75.

<prompt_4.1> Verify the steps taken to solve the equation and ensure the correct order of operations was followed. </prompt_4.1>

Let me plug the values back into the original equation to check.

<prompt_4.2> Substitute a = 2 and a = 8 back into the original equation to verify the solutions. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Then, 75 - [4*(5 - a)²]/3 = 63. Subtract 75: -[4*(5 - a)²]/3 = -12. Multiply both sides by -1: [4*(5 - a)²]/3 = 12. Multiply both sides by 3: 4*(5 - a)² = 36. Divide by 4: (5 - a)² = 9. Take square roots: 5 - a = ±3. Solve for a: a = 5 ±3. So, a = 8 or 2.
</execution_4.1>

<execution_4.2>
First, a = 2. Compute (5 - a)²: (5 - 2)² = 3² = 9. Then, 4*9 = 36. Divided by 3 is 12. Then, 75 - 12 = 63, which matches the right side.

Now, a = 8. Compute (5 - 8)² = (-3)² = 9. Then, 4*9 = 36. Divided by 3 is 12. Then, 75 - 12 = 63, which also matches.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.2:
I think that's all. The key steps were simplifying the equation step by step, isolating the squared term, taking the square root remembering to consider both positive and negative roots, solving the resulting linear equations, and then verifying the solutions by plugging them back into the original equation.
Everything checks out. So the sum is indeed 10.
</Plan_5>

<Execution_5>
No prompt to execute.
</Execution_5>

<Final_answer>
Therefore, the sum of the values of $a$ that satisfy the given equation is $\boxed{10}$.
</Final_answer>
</think>