## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Paul', 'hours worked by Mary', 'hours worked by John', 'hours worked by Laura']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $3x_0 + 6x_1 + 1x_2 + 5x_3$.

## Step 3: List the constraints
1. $3x_0 \geq 3$ is not needed as it's a fixed value, but $x_0$ has a likelihood to quit index of 3.
2. $27x_0$ is not needed as it's a fixed value, but $x_0$ has an organization score of 27.
3. $21x_1$ is not needed as it's a fixed value, but $x_1$ has a likelihood to quit index of 21.
4. $21x_1$ is not needed as it's a fixed value, but $x_1$ has an organization score of 21.
5. $8x_2$ is not needed as it's a fixed value, but $x_2$ has a likelihood to quit index of 8.
6. $26x_2$ is not needed as it's a fixed value, but $x_2$ has an organization score of 26.
7. $29x_3$ is not needed as it's a fixed value, but $x_3$ has a likelihood to quit index of 29.
8. $18x_3$ is not needed as it's a fixed value, but $x_3$ has an organization score of 18.
9. $3x_0 + 29x_3 \geq 25$
10. $21x_1 + 8x_2 \geq 43$
11. $3x_0 + 8x_2 \geq 44$
12. $3x_0 + 21x_1 + 29x_3 \geq 41$
13. $3x_0 + 21x_1 + 8x_2 \geq 41$
14. $21x_1 + 8x_2 + 29x_3 \geq 41$
15. $3x_0 + 21x_1 + 29x_3 \geq 47$
16. $3x_0 + 21x_1 + 8x_2 \geq 47$
17. $21x_1 + 8x_2 + 29x_3 \geq 47$
18. $3x_0 + 21x_1 + 29x_3 \geq 52$
19. $3x_0 + 21x_1 + 8x_2 \geq 52$
20. $21x_1 + 8x_2 + 29x_3 \geq 52$
21. $3x_0 + 21x_1 + 8x_2 + 29x_3 \geq 52$
22. $27x_0 + 21x_1 \geq 49$
23. $27x_0 + 18x_3 \geq 56$
24. $27x_0 + 21x_1 + 26x_2 + 18x_3 \geq 56$
25. $-9x_2 + 7x_3 \geq 0$
26. $-2x_0 + 1x_3 \geq 0$
27. $3x_0 + 21x_1 + 8x_2 \leq 104$
28. $21x_1 + 8x_2 + 29x_3 \leq 105$
29. $3x_0 + 21x_1 + 29x_3 \leq 128$
30. $27x_0 + 26x_2 \leq 331$
31. $27x_0 + 21x_1 + 18x_3 \leq 364$

## Step 4: Determine variable types
- $x_0$ (hours worked by Paul) can be non-integer.
- $x_1$ (hours worked by Mary) must be an integer.
- $x_2$ (hours worked by John) must be an integer.
- $x_3$ (hours worked by Laura) must be an integer.

## 5: Formulate the problem in Gurobi

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(lb=0, name="hours_worked_by_Paul")  # Can be non-integer
x1 = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="hours_worked_by_Mary")  # Must be integer
x2 = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="hours_worked_by_John")  # Must be integer
x3 = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="hours_worked_by_Laura")  # Must be integer

# Define the objective function
m.setObjective(3*x0 + 6*x1 + x2 + 5*x3, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x0 + 29*x3 >= 25)
m.addConstr(21*x1 + 8*x2 >= 43)
m.addConstr(3*x0 + 8*x2 >= 44)
m.addConstr(3*x0 + 21*x1 + 29*x3 >= 41)
m.addConstr(3*x0 + 21*x1 + 8*x2 >= 41)
m.addConstr(21*x1 + 8*x2 + 29*x3 >= 41)
m.addConstr(3*x0 + 21*x1 + 29*x3 >= 47)
m.addConstr(3*x0 + 21*x1 + 8*x2 >= 47)
m.addConstr(21*x1 + 8*x2 + 29*x3 >= 47)
m.addConstr(3*x0 + 21*x1 + 29*x3 >= 52)
m.addConstr(3*x0 + 21*x1 + 8*x2 >= 52)
m.addConstr(21*x1 + 8*x2 + 29*x3 >= 52)
m.addConstr(3*x0 + 21*x1 + 8*x2 + 29*x3 >= 52)
m.addConstr(27*x0 + 21*x1 >= 49)
m.addConstr(27*x0 + 18*x3 >= 56)
m.addConstr(27*x0 + 21*x1 + 26*x2 + 18*x3 >= 56)
m.addConstr(-9*x2 + 7*x3 >= 0)
m.addConstr(-2*x0 + x3 >= 0)
m.addConstr(3*x0 + 21*x1 + 8*x2 <= 104)
m.addConstr(21*x1 + 8*x2 + 29*x3 <= 105)
m.addConstr(3*x0 + 21*x1 + 29*x3 <= 128)
m.addConstr(27*x0 + 26*x2 <= 331)
m.addConstr(27*x0 + 21*x1 + 18*x3 <= 364)

# Solve the model
m.optimize()

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

```json
{
    'sym_variables': [
        ['x0', 'hours worked by Paul'], 
        ['x1', 'hours worked by Mary'], 
        ['x2', 'hours worked by John'], 
        ['x3', 'hours worked by Laura']
    ], 
    'objective_function': '3*x0 + 6*x1 + x2 + 5*x3', 
    'constraints': [
        '3*x0 + 29*x3 >= 25',
        '21*x1 + 8*x2 >= 43',
        '3*x0 + 8*x2 >= 44',
        '3*x0 + 21*x1 + 29*x3 >= 41',
        '3*x0 + 21*x1 + 8*x2 >= 41',
        '21*x1 + 8*x2 + 29*x3 >= 41',
        '3*x0 + 21*x1 + 29*x3 >= 47',
        '3*x0 + 21*x1 + 8*x2 >= 47',
        '21*x1 + 8*x2 + 29*x3 >= 47',
        '3*x0 + 21*x1 + 29*x3 >= 52',
        '3*x0 + 21*x1 + 8*x2 >= 52',
        '21*x1 + 8*x2 + 29*x3 >= 52',
        '3*x0 + 21*x1 + 8*x2 + 29*x3 >= 52',
        '27*x0 + 21*x1 >= 49',
        '27*x0 + 18*x3 >= 56',
        '27*x0 + 21*x1 + 26*x2 + 18*x3 >= 56',
        '-9*x2 + 7*x3 >= 0',
        '-2*x0 + x3 >= 0',
        '3*x0 + 21*x1 + 8*x2 <= 104',
        '21*x1 + 8*x2 + 29*x3 <= 105',
        '3*x0 + 21*x1 + 29*x3 <= 128',
        '27*x0 + 26*x2 <= 331',
        '27*x0 + 21*x1 + 18*x3 <= 364'
    ]
}
```