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

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $6x_0 + 7x_1 + 5x_2 + 3x_3$.

## Step 3: List all the constraints
1. $14x_0 + 12x_3 \geq 26$
2. $4x_1 + 13x_2 \geq 15$
3. $14x_0 + 13x_2 + 12x_3 \geq 15$
4. $4x_1 + 13x_2 + 10x_3 \geq 15$
5. $14x_0 + 4x_1 + 13x_2 \geq 15$
6. $14x_0 + 13x_2 + 12x_3 \geq 14$
7. $4x_1 + 13x_2 + 10x_3 \geq 14$
8. $14x_0 + 4x_1 + 13x_2 \geq 14$
9. $14x_0 + 13x_2 + 12x_3 \geq 20$
10. $4x_1 + 13x_2 + 10x_3 \geq 20$
11. $14x_0 + 4x_1 + 13x_2 \geq 20$
12. $17x_2 + 10x_3 \geq 39$
13. $8x_0 + 17x_2 + 10x_3 \geq 41$
14. $13x_2 + 12x_3 \leq 93$
15. $4x_1 + 12x_3 \leq 87$
16. $14x_0 + 4x_1 \leq 48$
17. $14x_0 + 12x_3 \leq 49$
18. $14x_0 + 13x_2 \leq 29$
19. $14x_0 + 4x_1 + 12x_3 \leq 111$
20. $14x_0 + 13x_2 + 12x_3 \leq 73$
21. $4x_1 + 13x_2 + 10x_3 \leq 62$
22. $14x_0 + 4x_1 + 13x_2 \leq 79$
23. $14x_0 + 4x_1 + 13x_2 + 12x_3 \leq 79$
24. $17x_2 + 10x_3 \leq 80$
25. $8x_0 + 17x_2 \leq 164$
26. $8x_0 + 13x_1 \leq 87$
27. $13x_1 + 17x_2 \leq 217$
28. $8x_0 + 13x_1 + 17x_2 \leq 136$
29. $8x_0 + 13x_1 + 10x_3 \leq 247$
30. $13x_1 + 17x_2 + 10x_3 \leq 88$
31. $8x_0 + 13x_1 + 17x_2 + 10x_3 \leq 88$
32. $11x_0 + 18x_1 \leq 68$
33. $11x_0 + 13x_3 \leq 46$
34. $11x_0 + 18x_1 + 18x_2 + 13x_3 \leq 46$

## Step 4: Convert the problem into Gurobi code
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", vtype=gp.GRB.INTEGER)  # hours worked by Hank
x1 = m.addVar(name="x1", vtype=gp.GRB.INTEGER)  # hours worked by Paul
x2 = m.addVar(name="x2", vtype=gp.GRB.INTEGER)  # hours worked by Laura
x3 = m.addVar(name="x3", vtype=gp.GRB.INTEGER)  # hours worked by John

# Define the objective function
m.setObjective(6 * x0 + 7 * x1 + 5 * x2 + 3 * x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(14 * x0 + 12 * x3 >= 26)
m.addConstr(4 * x1 + 13 * x2 >= 15)
m.addConstr(14 * x0 + 13 * x2 + 12 * x3 >= 15)
m.addConstr(4 * x1 + 13 * x2 + 10 * x3 >= 15)
m.addConstr(14 * x0 + 4 * x1 + 13 * x2 >= 15)
m.addConstr(14 * x0 + 13 * x2 + 12 * x3 >= 14)
m.addConstr(4 * x1 + 13 * x2 + 10 * x3 >= 14)
m.addConstr(14 * x0 + 4 * x1 + 13 * x2 >= 14)
m.addConstr(14 * x0 + 13 * x2 + 12 * x3 >= 20)
m.addConstr(4 * x1 + 13 * x2 + 10 * x3 >= 20)
m.addConstr(14 * x0 + 4 * x1 + 13 * x2 >= 20)
m.addConstr(17 * x2 + 10 * x3 >= 39)
m.addConstr(8 * x0 + 17 * x2 + 10 * x3 >= 41)
m.addConstr(13 * x2 + 12 * x3 <= 93)
m.addConstr(4 * x1 + 12 * x3 <= 87)
m.addConstr(14 * x0 + 4 * x1 <= 48)
m.addConstr(14 * x0 + 12 * x3 <= 49)
m.addConstr(14 * x0 + 13 * x2 <= 29)
m.addConstr(14 * x0 + 4 * x1 + 12 * x3 <= 111)
m.addConstr(14 * x0 + 13 * x2 + 12 * x3 <= 73)
m.addConstr(4 * x1 + 13 * x2 + 10 * x3 <= 62)
m.addConstr(14 * x0 + 4 * x1 + 13 * x2 <= 79)
m.addConstr(14 * x0 + 4 * x1 + 13 * x2 + 12 * x3 <= 79)
m.addConstr(17 * x2 + 10 * x3 <= 80)
m.addConstr(8 * x0 + 17 * x2 <= 164)
m.addConstr(8 * x0 + 13 * x1 <= 87)
m.addConstr(13 * x1 + 17 * x2 <= 217)
m.addConstr(8 * x0 + 13 * x1 + 17 * x2 <= 136)
m.addConstr(8 * x0 + 13 * x1 + 10 * x3 <= 247)
m.addConstr(13 * x1 + 17 * x2 + 10 * x3 <= 88)
m.addConstr(8 * x0 + 13 * x1 + 17 * x2 + 10 * x3 <= 88)
m.addConstr(11 * x0 + 18 * x1 <= 68)
m.addConstr(11 * x0 + 13 * x3 <= 46)
m.addConstr(11 * x0 + 18 * x1 + 18 * x2 + 13 * x3 <= 46)

# Solve the model
m.optimize()

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

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'), 
        ('x1', 'hours worked by Paul'), 
        ('x2', 'hours worked by Laura'), 
        ('x3', 'hours worked by John')
    ], 
    'objective_function': '6*x0 + 7*x1 + 5*x2 + 3*x3', 
    'constraints': [
        '14*x0 + 12*x3 >= 26', 
        '4*x1 + 13*x2 >= 15', 
        '14*x0 + 13*x2 + 12*x3 >= 15', 
        '4*x1 + 13*x2 + 10*x3 >= 15', 
        '14*x0 + 4*x1 + 13*x2 >= 15', 
        '14*x0 + 13*x2 + 12*x3 >= 14', 
        '4*x1 + 13*x2 + 10*x3 >= 14', 
        '14*x0 + 4*x1 + 13*x2 >= 14', 
        '14*x0 + 13*x2 + 12*x3 >= 20', 
        '4*x1 + 13*x2 + 10*x3 >= 20', 
        '14*x0 + 4*x1 + 13*x2 >= 20', 
        '17*x2 + 10*x3 >= 39', 
        '8*x0 + 17*x2 + 10*x3 >= 41', 
        '13*x2 + 12*x3 <= 93', 
        '4*x1 + 12*x3 <= 87', 
        '14*x0 + 4*x1 <= 48', 
        '14*x0 + 12*x3 <= 49', 
        '14*x0 + 13*x2 <= 29', 
        '14*x0 + 4*x1 + 12*x3 <= 111', 
        '14*x0 + 13*x2 + 12*x3 <= 73', 
        '4*x1 + 13*x2 + 10*x3 <= 62', 
        '14*x0 + 4*x1 + 13*x2 <= 79', 
        '14*x0 + 4*x1 + 13*x2 + 12*x3 <= 79', 
        '17*x2 + 10*x3 <= 80', 
        '8*x0 + 17*x2 <= 164', 
        '8*x0 + 13*x1 <= 87', 
        '13*x1 + 17*x2 <= 217', 
        '8*x0 + 13*x1 + 17*x2 <= 136', 
        '8*x0 + 13*x1 + 10*x3 <= 247', 
        '13*x1 + 17*x2 + 10*x3 <= 88', 
        '8*x0 + 13*x1 + 17*x2 + 10*x3 <= 88', 
        '11*x0 + 18*x1 <= 68', 
        '11*x0 + 13*x3 <= 46', 
        '11*x0 + 18*x1 + 18*x2 + 13*x3 <= 46'
    ]
}
```