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:

Factuality: Did the model extract pertinent and accurate background knowledge without any misinformation when factual knowledge retrieval is needed? Is the response supported by reliable evidence or citation of the source of its information?
Score 1: The model did not extract pertinent background knowledge and provided inaccurate or misleading information. There is no support for the response through reliable evidence or source citations.
Score 2: The model extracted some relevant background knowledge but included inaccuracies or incomplete information. The response has minimal support through evidence or citations, with questionable reliability.
Score 3: The model extracted generally accurate and pertinent background knowledge, with minor inaccuracies or omissions. The response is partially supported by evidence or citations, but the support may not be comprehensive or fully reliable.
Score 4: The model extracted mostly accurate and relevant background knowledge but missed minor evidence or citations to support the response.
Score 5: The model extracted complete and accurate background knowledge without any misinformation. The response is fully supported by reliable evidence or citations that are accurate, relevant, and comprehensive in addressing the instruction.
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 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]
A polynomial with integer coefficients is of the form \[2x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1 = 0.\]Find the number of different possible rational roots of this polynomial.

[Ground truth Answer]
By the Rational Root Theorem, the only possible rational roots are of the form $\pm \frac{a}{b},$ where $a$ divides 1 and $b$ divides 2. Thus, the possible rational roots are \[\pm 1, \ \pm \frac{1}{2}.\]Thus, there are $\boxed{4}$ possible rational roots.

[Assistant's Response]
To find the number of different possible rational roots of the polynomial 

\[
2x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1 = 0,
\]

we can apply the Rational Root Theorem. This theorem states that any rational root, expressed in the form \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.

In this polynomial:

- The leading coefficient is \(2\).
- The constant term is \(1\).

**Step 1: Identify the factors of the constant term and the leading coefficient.**

1. **Factors of the constant term (1)**: The factors of \(1\) are \(\pm 1\).
2. **Factors of the leading coefficient (2)**: The factors of \(2\) are \(\pm 1\) and \(\pm 2\).

**Step 2: List the possible rational roots \(\frac{p}{q}\)** where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

- For \(p = 1\):
  - \(\frac{1}{1} = 1\)
  - \(\frac{1}{2} = \frac{1}{2}\)

- For \(p = -1\):
  - \(\frac{-1}{1} = -1\)
  - \(\frac{-1}{2} = -\frac{1}{2}\)

**Step 3: Compile the list of possible rational roots.**

The possible rational roots are:
- \(1\)
- \(\frac{1}{2}\)
- \(-1\)
- \(-\frac{1}{2}\)

**Step 4: Count the different possible rational roots.**

We have the following distinct rational roots:
1. \(1\)
2. \(\frac{1}{2}\)
3. \(-1\)
4. \(-\frac{1}{2}\)

Thus, the total number of different possible rational roots is 

\[
\boxed{4}.
\]

[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: