### Component 1 (Lines 1-3)
- **Description:** Introduction to the problem and identification of the task.
- **Plan:** Lines 1-3
- **Prompt:** Introduce the problem and identify the task we need to solve.
- **Execution:** Lines not found
- **Comment:** Lines not found

### Component 2 (Lines 4-7)
- **Description:** Initial thoughts and planning the approach to solve the problem.
- **Plan:** Lines 4-7
- **Prompt:** Think aloud about the problem and plan the approach to solve it.
- **Execution:** Lines not found
- **Comment:** Lines not found

### Component 3 (Lines 8-10)
- **Description:** Recalling the Conjugate Root Theorem and planning to apply it.
- **Plan:** Lines 8-10
- **Prompt:** Recall the Conjugate Root Theorem and plan to apply it to the given roots.
- **Execution:** Lines not found
- **Comment:** Lines not found

### Component 4 (Lines 11-22)
- **Description:** Identifying the conjugates of the given roots and determining the total number of roots needed.
- **Plan:** Lines 11-13
- **Prompt:** Identify the conjugates of the given roots and determine if they are already included or need to be added.
- **Execution:** Lines 14-22
- **Comment:** Lines not found

### Component 5 (Lines 23-31)
- **Description:** Re-evaluating the count of roots and determining the degree of the polynomial.
- **Plan:** Lines 23-24
- **Prompt:** Re-evaluate the count of roots and determine the degree of the polynomial.
- **Execution:** Lines 25-31
- **Comment:** Lines not found

### Component 6 (Lines 32-34)
- **Description:** Considering alternative methods to find the minimal polynomial.
- **Plan:** Lines 32-34
- **Prompt:** Consider alternative methods to find the minimal polynomial.
- **Execution:** Lines not found
- **Comment:** Lines not found

### Component 7 (Lines 35-44)
- **Description:** Constructing the polynomial step by step using the identified roots.
- **Plan:** Lines 35-36
- **Prompt:** Construct the polynomial step by step using the identified roots.
- **Execution:** Lines 37-44
- **Comment:** Lines not found

### Component 8 (Lines 45-51)
- **Description:** Verifying the degree of the polynomial and resolving contradictions.
- **Plan:** Lines 45-46
- **Prompt:** Verify the degree of the polynomial and resolve any contradictions.
- **Execution:** Lines 47-51
- **Comment:** Lines not found

### Component 9 (Lines 52-56)
- **Description:** Finalizing the minimal polynomial and confirming its degree.
- **Plan:** Lines 52-53
- **Prompt:** Finalize the minimal polynomial and confirm its degree.
- **Execution:** Lines 54-56
- **Comment:** Lines not found

### Component 10 (Lines 57-72)
- **Description:** Exploring potential dependencies or relations between roots for a lower degree polynomial.
- **Plan:** Lines 57-58
- **Prompt:** Explore potential dependencies or relations between roots for a lower degree polynomial.
- **Execution:** Lines 59-72
- **Comment:** Lines not found

### Component 11 (Lines 73-81)
- **Description:** Numerical verification of distinct roots.
- **Plan:** Lines 73-74
- **Prompt:** Perform numerical verification to check if the roots are distinct.
- **Execution:** Lines 75-81
- **Comment:** Lines not found

### Component 12 (Lines 82-85)
- **Description:** Confirming the distinctness and irreducibility of quadratics.
- **Plan:** Lines 82-83
- **Prompt:** Confirm the distinctness and irreducibility of the quadratics.
- **Execution:** Lines 84-85
- **Comment:** Lines not found

### Component 13 (Lines 86-100)
- **Description:** Further exploration of algebraic relations and confirming the minimal degree.
- **Plan:** Lines 86-87
- **Prompt:** Explore further algebraic relations and confirm the minimal degree.
- **Execution:** Lines 88-100
- **Comment:** Lines not found

### Component 14 (Lines 101-109)
- **Description:** Final verification and conclusion of the solution.
- **Plan:** Lines 101-102
- **Prompt:** Perform a final verification and conclude the solution.
- **Execution:** Lines 103-109
- **Comment:** Lines 110

### Component 15 (Lines 111-115)
- **Description:** Final check for irreducibility of quadratics and confirmation of the solution.
- **Plan:** Lines 111-112
- **Prompt:** Check for irreducibility of quadratics and confirm the solution.
- **Execution:** Lines 113-115
- **Comment:** Lines not found

### Component 16 (Lines 116-117)
- **Description:** Presenting the final answer.
- **Plan:** Lines 116
- **Prompt:** Present the final answer.
- **Execution:** Lines 117
- **Comment:** Lines not found