System: You are a helpful and precise assistant in checking the quality of the answer.
User: We would like to request your feedback on the performance of the response of the assistant to the user instruction displayed below. In the feedback, I want you to rate the quality of the response in these 3 categories according to each scoring rubric:

Logical Correctness: Is the final answer provided by the response logically accurate and correct for an instruction that has a deterministic answer?
Score 1: The model's final answer is completely incorrect and lacks sound reasoning.
Score 2: The model's final answer contains significant errors that critically undermine its correctness.
Score 3: The model's final answer includes inaccuracies that require considerable effort to correct.
Score 4: The model's final answer contains minor errors, which are easy to rectify and do not significantly impact its overall correctness.
Score 5: The model's final answer is completely accurate and sound.
Logical Efficiency: Is the response logically efficient? The logic behind the response should have no redundant step, remaining simple and efficient. For tasks involving coding, the proposed solution should also consider time complexity.
Score 1: The logic behind the response is significantly inefficient and redundant, necessitating a complete reorganization of logic for clarity and efficiency.
Score 2: The logic of the response lacks efficiency and conciseness, requiring a substantial reorganization for better optimization.
Score 3: The logic of the response is not efficient enough, necessitating major edits for improved optimization.
Score 4: The logic of the response is largely efficient, but it still has some redundant steps. It could be handled from minor edits for better optimization.
Score 5: The logic of the response is optimally efficient, requiring no further optimization.
Logical Robustness: Does the model ensure general applicability and avoid logical contradictions in its reasoning steps for an instruction that requires step-by-step logical process? This includes the consideration of edge cases for coding and mathematical problems, and the absence of any counterexamples.
Score 1: The logic of the model's response is completely incoherent.
Score 2: The model's response contains major logical inconsistencies or errors.
Score 3: The model's response contains some logical inconsistencies or errors, but they are not significant.
Score 4: The model's response is logically sound, but it does not consider some edge cases.
Score 5: The model's response is logically flawless and it takes into account all potential edge cases.
[Instruction]
Simplify\n\[\cos \left( \frac{2 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right).\]

[Ground truth Answer]
Let $x = \cos \left( \frac{2 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right).$ Then by repeated application of the double angle formula,\n\begin{align*}\nx \sin \left( \frac{2 \pi}{15} \right) &= \sin \left( \frac{2 \pi}{15} \right) \cos \left( \frac{2 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\\n&= \frac{1}{2} \sin \left( \frac{4 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\\n&= \frac{1}{4} \sin \left (\frac {8 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\\n&= \frac{1}{8} \sin \left( \frac{16 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\\n&= \frac{1}{16} \sin \left( \frac{32 \pi}{15} \right) \\\n&= \frac{1}{16} \sin \left( \frac{2 \pi}{15} \right),\n\end{align*}so $x = \boxed{\frac{1}{16}}.$

[Assistant's Response]
To simplify the expression 

\[
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac{16 \pi}{15} \right),
\]

we can utilize the property of the cosine function, particularly that \(\cos(x) = -\cos(\pi - x)\). 

First, we note that:

\[
\cos \left( \frac{16 \pi}{15} \right) = \cos \left( 2\pi - \frac{16 \pi}{15} \right) = \cos \left( \frac{14 \pi}{15} \right) = -\cos \left( \frac{\pi}{15} \right).
\]

This allows us to rewrite the original expression as:

\[
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) (-\cos \left( \frac{\pi}{15} \right)).
\]

Next, we can pair the cosines for further simplification. Specifically, we pair \(\cos \left( \frac{2 \pi}{15} \right)\) and \(\cos \left( \frac{8 \pi}{15} \right)\):

\[
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) = \frac{1}{2} \left( \cos \left( \frac{6 \pi}{15} \right) + \cos \left( \frac{10 \pi}{15} \right) \right) = \frac{1}{2} \left( \cos \left( \frac{2 \pi}{5} \right) + \cos \left( \frac{2 \pi}{3} \right) \right).
\]

We know that \(\cos \left( \frac{2 \pi}{3} \right) = -\frac{1}{2}\). Thus, we can substitute this value in:

\[
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) = \frac{1}{2} \left( \cos \left( \frac{2 \pi}{5} \right) - \frac{1}{2} \right).
\]

Now, substituting back into our expression, we have:

\[
-\frac{1}{2} \left( \cos \left( \frac{2 \pi}{5} \right) - \frac{1}{2} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{\pi}{15} \right).
\]

This expression can be simplified further, but the key takeaway is that the original expression can be reduced to a product involving known cosine values. 

In conclusion, the simplified expression is:

\[
-\frac{1}{2} \left( \cos \left( \frac{2 \pi}{5} \right) - \frac{1}{2} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{\pi}{15} \right).
\]

This showcases how the cosine values can be manipulated to simplify the overall expression. Further numerical evaluation may be necessary depending on the context or required precision.

[The End of Assistant's Response]

Please give feedback on the assistant's responses. Also, provide the assistant with a score on a scale of 1 to 5 for each category, where a higher score indicates better overall performance. Make sure to give feedback or comments for each category first and then write the score for each category. Only write the feedback corresponding to the scoring rubric for each category. The scores of each category should be orthogonal, indicating that 'Efficiency of User Alignment' should not be considered for 'Readability of User Alignment' category, for example. 

Lastly, return a Python dictionary object that has skillset names as keys and the corresponding scores as values.

[System]



Assistant: