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

## Step 2: Convert the given problem into a symbolic representation
The objective function to minimize is $6x_0 + 6x_1 + 4x_2$.

## 3: List all the constraints
The constraints given are:
- $7x_1 + 3x_2 \geq 35$
- $13x_0 + 7x_1 \geq 28$
- $13x_0 + 7x_1 + 3x_2 \geq 28$
- $8x_0 + 7x_1 \geq 29$
- $8x_0 + 2x_2 \geq 13$
- $8x_0 + 7x_1 + 2x_2 \geq 17$
- $8x_0 + 7x_1 + 2x_2 \geq 17$
- $13x_0 + 6x_2 \geq 10$
- $13x_1 + 6x_2 \geq 19$
- $13x_0 + 13x_1 \geq 12$
- $13x_0 + 13x_1 + 6x_2 \geq 12$
- $8x_1 + 3x_2 \geq 24$
- $2x_0 + 8x_1 \geq 21$
- $2x_0 + 3x_2 \geq 23$
- $2x_0 + 8x_1 + 3x_2 \geq 23$
- $12x_0 + 6x_1 \geq 29$
- $12x_0 + 14x_2 \geq 27$
- $6x_1 + 14x_2 \geq 37$
- $12x_0 + 6x_1 + 14x_2 \geq 40$
- $12x_0 + 6x_1 + 14x_2 \geq 40$
- $-10x_0 + 5x_2 \geq 0$
- $8x_0 + 7x_1 + 2x_2 \leq 68$
- $2x_0 + 3x_2 \leq 62$
- $8x_1 + 3x_2 \leq 72$
- $2x_0 + 8x_1 \leq 47$

## 4: Define the symbolic representation of the problem
The symbolic representation is:
```json
{
    'sym_variables': [
        ('x0', 'hours worked by John'),
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by Hank')
    ],
    'objective_function': '6*x0 + 6*x1 + 4*x2',
    'constraints': [
        '7*x1 + 3*x2 >= 35',
        '13*x0 + 7*x1 >= 28',
        '13*x0 + 7*x1 + 3*x2 >= 28',
        '8*x0 + 7*x1 >= 29',
        '8*x0 + 2*x2 >= 13',
        '8*x0 + 7*x1 + 2*x2 >= 17',
        '8*x0 + 7*x1 + 2*x2 >= 17',
        '13*x0 + 6*x2 >= 10',
        '13*x1 + 6*x2 >= 19',
        '13*x0 + 13*x1 >= 12',
        '13*x0 + 13*x1 + 6*x2 >= 12',
        '8*x1 + 3*x2 >= 24',
        '2*x0 + 8*x1 >= 21',
        '2*x0 + 3*x2 >= 23',
        '2*x0 + 8*x1 + 3*x2 >= 23',
        '12*x0 + 6*x1 >= 29',
        '12*x0 + 14*x2 >= 27',
        '6*x1 + 14*x2 >= 37',
        '12*x0 + 6*x1 + 14*x2 >= 40',
        '12*x0 + 6*x1 + 14*x2 >= 40',
        '-10*x0 + 5*x2 >= 0',
        '8*x0 + 7*x1 + 2*x2 <= 68',
        '2*x0 + 3*x2 <= 62',
        '8*x1 + 3*x2 <= 72',
        '2*x0 + 8*x1 <= 47',
        'x0 % 1 == 0',
        'x1 % 1 == 0',
        'x2 % 1 == 0'
    ]
}
```

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

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name='x0', vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x1 = model.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Laura
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Hank

    # Objective function
    model.setObjective(6 * x0 + 6 * x1 + 4 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(7 * x1 + 3 * x2 >= 35)
    model.addConstr(13 * x0 + 7 * x1 >= 28)
    model.addConstr(13 * x0 + 7 * x1 + 3 * x2 >= 28)
    model.addConstr(8 * x0 + 7 * x1 >= 29)
    model.addConstr(8 * x0 + 2 * x2 >= 13)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x2 >= 17)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x2 >= 17)
    model.addConstr(13 * x0 + 6 * x2 >= 10)
    model.addConstr(13 * x1 + 6 * x2 >= 19)
    model.addConstr(13 * x0 + 13 * x1 >= 12)
    model.addConstr(13 * x0 + 13 * x1 + 6 * x2 >= 12)
    model.addConstr(8 * x1 + 3 * x2 >= 24)
    model.addConstr(2 * x0 + 8 * x1 >= 21)
    model.addConstr(2 * x0 + 3 * x2 >= 23)
    model.addConstr(2 * x0 + 8 * x1 + 3 * x2 >= 23)
    model.addConstr(12 * x0 + 6 * x1 >= 29)
    model.addConstr(12 * x0 + 14 * x2 >= 27)
    model.addConstr(6 * x1 + 14 * x2 >= 37)
    model.addConstr(12 * x0 + 6 * x1 + 14 * x2 >= 40)
    model.addConstr(12 * x0 + 6 * x1 + 14 * x2 >= 40)
    model.addConstr(-10 * x0 + 5 * x2 >= 0)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x2 <= 68)
    model.addConstr(2 * x0 + 3 * x2 <= 62)
    model.addConstr(8 * x1 + 3 * x2 <= 72)
    model.addConstr(2 * x0 + 8 * x1 <= 47)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('x0: ', x0.varValue)
        print('x1: ', x1.varValue)
        print('x2: ', x2.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```