## Step 1: Define the variables and their corresponding natural language objects
Let's denote the hours worked by Hank, Mary, Paul, and Ringo as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $6x_0^2 + 3x_0x_1 + 6x_1x_3 + 6x_2x_3 + 8x_3^2 + x_0 + 8x_1$.

## 3: List the constraints in symbolic notation
Constraints:
1. $4x_0 \leq 113$
2. $13x_0 \leq 134$
3. $x_0 \leq 133$
4. $2x_1 \leq 113$
5. $10x_1 \leq 134$
6. $x_1 \leq 133$
7. $12x_2 \leq 113$
8. $14x_2 \leq 134$
9. $14x_2 \leq 133$
10. $12x_3 \leq 113$
11. $6x_3 \leq 134$
12. $11x_3 \leq 133$
13. $4x_0^2 + 12x_2^2 \geq 26$
14. $2x_1 + 12x_3 \geq 12$
15. $12x_2 + 12x_3 \geq 20$
16. $2x_1 + 12x_2 \geq 26$
17. $4x_0 + 12x_3 \geq 11$
18. $4x_0^2 + 4x_1^2 + 12x_3^2 \geq 20$
19. $4x_1^2 + 12x_2^2 + 12x_3^2 \geq 20$
20. $4x_0 + 12x_2 + 12x_3 \geq 20$
21. $4x_0 + 2x_1 + 12x_2 \geq 20$
22. $4x_0^2 + 4x_1^2 + 12x_3^2 \geq 17$
23. $4x_1^2 + 12x_2^2 + 12x_3^2 \geq 17$
24. $4x_0 + 12x_2 + 12x_3 \geq 17$
25. $4x_0 + 2x_1 + 12x_2 \geq 17$
26. $4x_0^2 + 4x_1^2 + 12x_3^2 \geq 15$
27. $2x_1 + 12x_2 + 12x_3 \geq 15$
28. $4x_0 + 12x_2 + 12x_3 \geq 15$
29. $4x_0 + 2x_1 + 12x_2 \geq 15$
30. $4x_0^2 + 4x_1^2 + 4x_2^2 \geq 15$
31. $13x_0^2 + 14x_2^2 \geq 13$
32. $13x_0 + 10x_1 + 6x_3 \geq 20$
33. $x_0^2 + x_1^2 \geq 12$
34. $14x_2^2 + 11x_3^2 \geq 33$
35. $x_0 + 6x_3 \geq 19$
36. $x_0 + x_2 \geq 11$
37. $x_1 + 6x_3 \geq 16$
38. $2x_1 + 12x_3 \leq 89$
39. $4x_0 + 12x_2 \leq 88$
40. $2x_1 + 12x_2 \leq 36$
41. $4x_0 + 2x_1 + 12x_2 + 12x_3 \leq 36$
42. $13x_0 + 6x_3 \leq 81$
43. $10x_1^2 + 14x_2^2 \leq 61$
44. $13x_0 + 14x_2 + 6x_3 \leq 51$
45. $13x_0 + 10x_1 + 14x_2 \leq 109$
46. $13x_0^2 + 10x_1^2 + 6x_3^2 \leq 78$
47. $13x_0 + 10x_1 + 14x_2 + 6x_3 \leq 78$
48. $x_1 + 14x_2 \leq 35$
49. $x_0 + 6x_3 \leq 131$
50. $x_1 + 6x_3 \leq 98$
51. $x_0 + x_1 \leq 127$
52. $x_0 + x_1 + 14x_2 \leq 129$
53. $x_0 + x_1 + 14x_2 + 6x_3 \leq 129$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'),
        ('x1', 'hours worked by Mary'),
        ('x2', 'hours worked by Paul'),
        ('x3', 'hours worked by Ringo')
    ],
    'objective_function': '6*x0^2 + 3*x0*x1 + 6*x1*x3 + 6*x2*x3 + 8*x3^2 + x0 + 8*x1',
    'constraints': [
        '4*x0 <= 113',
        '13*x0 <= 134',
        'x0 <= 133',
        '2*x1 <= 113',
        '10*x1 <= 134',
        'x1 <= 133',
        '12*x2 <= 113',
        '14*x2 <= 134',
        '14*x2 <= 133',
        '12*x3 <= 113',
        '6*x3 <= 134',
        '11*x3 <= 133',
        '4*x0^2 + 12*x2^2 >= 26',
        '2*x1 + 12*x3 >= 12',
        '12*x2 + 12*x3 >= 20',
        '2*x1 + 12*x2 >= 26',
        '4*x0 + 12*x3 >= 11',
        '4*x0^2 + 4*x1^2 + 12*x3^2 >= 20',
        '4*x1^2 + 12*x2^2 + 12*x3^2 >= 20',
        '4*x0 + 12*x2 + 12*x3 >= 20',
        '4*x0 + 2*x1 + 12*x2 >= 20',
        '4*x0^2 + 4*x1^2 + 12*x3^2 >= 17',
        '4*x1^2 + 12*x2^2 + 12*x3^2 >= 17',
        '4*x0 + 12*x2 + 12*x3 >= 17',
        '4*x0 + 2*x1 + 12*x2 >= 17',
        '4*x0^2 + 4*x1^2 + 12*x3^2 >= 15',
        '2*x1 + 12*x2 + 12*x3 >= 15',
        '4*x0 + 12*x2 + 12*x3 >= 15',
        '4*x0 + 2*x1 + 12*x2 >= 15',
        '4*x0^2 + 4*x1^2 + 4*x2^2 >= 15',
        '13*x0^2 + 14*x2^2 >= 13',
        '13*x0 + 10*x1 + 6*x3 >= 20',
        'x0^2 + x1^2 >= 12',
        '14*x2^2 + 11*x3^2 >= 33',
        'x0 + 6*x3 >= 19',
        'x0 + x2 >= 11',
        'x1 + 6*x3 >= 16',
        '2*x1 + 12*x3 <= 89',
        '4*x0 + 12*x2 <= 88',
        '2*x1 + 12*x2 <= 36',
        '4*x0 + 2*x1 + 12*x2 + 12*x3 <= 36',
        '13*x0 + 6*x3 <= 81',
        '10*x1^2 + 14*x2^2 <= 61',
        '13*x0 + 14*x2 + 6*x3 <= 51',
        '13*x0 + 10*x1 + 14*x2 <= 109',
        '13*x0^2 + 10*x1^2 + 6*x3^2 <= 78',
        '13*x0 + 10*x1 + 14*x2 + 6*x3 <= 78',
        'x1 + 14*x2 <= 35',
        'x0 + 6*x3 <= 131',
        'x1 + 6*x3 <= 98',
        'x0 + x1 <= 127',
        'x0 + x1 + 14*x2 <= 129',
        'x0 + x1 + 14*x2 + 6*x3 <= 129'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Hank
x1 = m.addVar(name="x1", lb=0)  # hours worked by Mary
x2 = m.addVar(name="x2", lb=0, integrality=1)  # hours worked by Paul
x3 = m.addVar(name="x3", lb=0)  # hours worked by Ringo

# Objective function
m.setObjective(6*x0**2 + 3*x0*x1 + 6*x1*x3 + 6*x2*x3 + 8*x3**2 + x0 + 8*x1, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(4*x0 <= 113)
m.addConstr(13*x0 <= 134)
m.addConstr(x0 <= 133)
m.addConstr(2*x1 <= 113)
m.addConstr(10*x1 <= 134)
m.addConstr(x1 <= 133)
m.addConstr(12*x2 <= 113)
m.addConstr(14*x2 <= 134)
m.addConstr(14*x2 <= 133)
m.addConstr(12*x3 <= 113)
m.addConstr(6*x3 <= 134)
m.addConstr(11*x3 <= 133)
m.addConstr(4*x0**2 + 12*x2**2 >= 26)
m.addConstr(2*x1 + 12*x3 >= 12)
m.addConstr(12*x2 + 12*x3 >= 20)
m.addConstr(2*x1 + 12*x2 >= 26)
m.addConstr(4*x0 + 12*x3 >= 11)
m.addConstr(4*x0**2 + 4*x1**2 + 12*x3**2 >= 20)
m.addConstr(4*x1**2 + 12*x2**2 + 12*x3**2 >= 20)
m.addConstr(4*x0 + 12*x2 + 12*x3 >= 20)
m.addConstr(4*x0 + 2*x1 + 12*x2 >= 20)
m.addConstr(4*x0**2 + 4*x1**2 + 12*x3**2 >= 17)
m.addConstr(4*x1**2 + 12*x2**2 + 12*x3**2 >= 17)
m.addConstr(4*x0 + 12*x2 + 12*x3 >= 17)
m.addConstr(4*x0 + 2*x1 + 12*x2 >= 17)
m.addConstr(4*x0**2 + 4*x1**2 + 12*x3**2 >= 15)
m.addConstr(2*x1 + 12*x2 + 12*x3 >= 15)
m.addConstr(4*x0 + 12*x2 + 12*x3 >= 15)
m.addConstr(4*x0 + 2*x1 + 12*x2 >= 15)
m.addConstr(4*x0**2 + 4*x1**2 + 4*x2**2 >= 15)
m.addConstr(13*x0**2 + 14*x2**2 >= 13)
m.addConstr(13*x0 + 10*x1 + 6*x3 >= 20)
m.addConstr(x0**2 + x1**2 >= 12)
m.addConstr(14*x2**2 + 11*x3**2 >= 33)
m.addConstr(x0 + 6*x3 >= 19)
m.addConstr(x0 + x2 >= 11)
m.addConstr(x1 + 6*x3 >= 16)
m.addConstr(2*x1 + 12*x3 <= 89)
m.addConstr(4*x0 + 12*x2 <= 88)
m.addConstr(2*x1 + 12*x2 <= 36)
m.addConstr(4*x0 + 2*x1 + 12*x2 + 12*x3 <= 36)
m.addConstr(13*x0 + 6*x3 <= 81)
m.addConstr(10*x1**2 + 14*x2**2 <= 61)
m.addConstr(13*x0 + 14*x2 + 6*x3 <= 51)
m.addConstr(13*x0 + 10*x1 + 14*x2 <= 109)
m.addConstr(13*x0**2 + 10*x1**2 + 6*x3**2 <= 78)
m.addConstr(13*x0 + 10*x1 + 14*x2 + 6*x3 <= 78)
m.addConstr(x1 + 14*x2 <= 35)
m.addConstr(x0 + 6*x3 <= 131)
m.addConstr(x1 + 6*x3 <= 98)
m.addConstr(x0 + x1 <= 127)
m.addConstr(x0 + x1 + 14*x2 <= 129)
m.addConstr(x0 + x1 + 14*x2 + 6*x3 <= 129)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Hank: ", x0.varValue)
    print("Hours worked by Mary: ", x1.varValue)
    print("Hours worked by Paul: ", x2.varValue)
    print("Hours worked by Ringo: ", x3.varValue)
else:
    print("The model is infeasible")
```