<Question>
The operation @ is defined as $\frac{m}{n}@\frac{p}{q} = (m)(p)\left(\frac{q}{n}\right)$ for simplified fractions $\frac{p}{q}$.  What is the simplified value of $\frac{7}{30}@\frac{10}{21}$?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out the value of this operation @ when applied to the fractions 7/30 and 10/21. The operation is defined as (m/n) @ (p/q) = m * p * (q/n), but they mentioned that the fractions need to be simplified. Hmm, let me parse this step by step.

First, let me make sure I understand the operation correctly.
The fractions given are 7/30 and 10/21. The operation @ takes two fractions, each in their simplified form, and then combines them using the formula (m/n) @ (p/q) = m * p * (q/n). Wait, but the fractions need to be simplified. So maybe before applying the operation, I need to ensure that both fractions are in their simplest forms. Let me check if 7/30 and 10/21 are already simplified.

Starting with 7/30: 7 is a prime number, and 30 is 2*3*5. Since 7 shares no common factors with 30, that's simplified.

Next, 10/21: 10 is 2*5, and 21 is 3*7. Again, no common factors, so that's also simplified.
Alright, so both fractions are already simplified.

Now, according to the operation's definition, (m/n) @ (p/q) = m * p * (q/n). Let's break down the components here. For the first fraction, m/n is 7/30, so m is 7 and n is 30. For the second fraction, p/q is 10/21, so p is 10 and q is 21. Plugging into the formula, we get:

<prompt_1.1> Apply the operation @ to the fractions 7/30 and 10/21 using the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
7 * 10 * (21/30)

Wait, let me confirm that. The formula is m * p * (q/n). Here, q is the denominator of the second fraction, which is 21, and n is the denominator of the first fraction, which is 30. So yes, it's 21 divided by 30.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
So first, compute 21 divided by 30. That simplifies to 7/10, right?
Because 21 divided by 3 is 7, and 30 divided by 3 is 10. So 21/30 reduces to 7/10.

Therefore, the operation becomes 7 * 10 * (7/10).

Let me compute that step by step. Multiply 7 and 10 first: 7*10=70. Then multiply that by 7/10.
So 70 * (7/10) = (70/10)*7 = 7*7 = 49.

Wait, is that right? Let me check again.

<prompt_2.1> Verify the calculation of the operation @ using the above plan. </prompt_2.1>

So the result is 49. But I need to make sure that this is in its simplest form.
49 is 7 squared, and 1 is just 1, so 49/1 is already simplified.

Therefore, the simplified value is 49/1, which is just 49.
-----

Wait, but let me go back through the steps again to ensure I didn't make any mistakes.

<prompt_2.2> Re-evaluate the steps of the operation to ensure accuracy using the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
7 * 10 is 70. Then 70 multiplied by 7/10. Alternatively, 70 * 7 = 490, and 490 divided by 10 is 49.
</execution_2.1>

<execution_2.2>
First, confirm the fractions are simplified. 7/30: 7 and 30 share no common factors. 10/21: 10 and 21 share no common factors. Correct.

Operation formula: m * p * (q/n). So m=7, n=30, p=10, q=21. So 7*10*(21/30). Compute 21/30 first. Simplify 21/30 by dividing numerator and denominator by 3: 7/10. Then 7*10*(7/10). Compute 7*10=70. Then 70*(7/10). Multiply 70 by 7: 490. Divide by 10: 49.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Alternatively, maybe there's another way to compute it. Let's try simplifying before multiplying.

<prompt_3.1> Explore an alternative method to compute the operation by simplifying first using the above plan. </prompt_3.1>
-----

Based on execution_2.2:
Another check: 7*10*21 / 30. So that's (7*10*21)/30.

<prompt_3.2> Perform another check of the operation by computing it as a single fraction using the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Looking at 7 * 10 * (21/30). Maybe we can cancel out some factors before multiplying everything together. Let's see. 10 and 30 have a common factor of 10. So 10 divided by 10 is 1, and 30 divided by 10 is 3. So then the expression becomes 7 * 1 * 21/3. Wait, 21/3 is 7. So then 7 * 1 * 7 = 49.
</execution_3.1>

<execution_3.2>
Let me compute numerator and denominator separately. Numerator: 7*10=70; 70*21=1470. Denominator: 30. So 1470/30. Divide numerator and denominator by 10: 147/3. Then 147 divided by 3 is 49.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But let me confirm once more. Let's compute 7*10*(21/30).
7*10 is 70. 70*(21/30) is (70*21)/30. 70 divided by 30 is 7/3. Then 7/3 *21 is (7*21)/3 = (147)/3 = 49.
-----

Based on execution_3.2:
Yes, that's 49. So regardless of the method, the answer comes out to 49.
Therefore, the simplified value is 49.
</Plan_4>

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

<Final_answer>
Thus, the simplified value of $\frac{7}{30}@\frac{10}{21}$ is $\boxed{\frac{49}{1}}$.
</Final_answer>
</think>