## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Paul', and 'hours worked by Mary', which can be represented as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $2x_0 + 7x_1 + 7x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $16x_0 \leq 169$
- $11x_0 \leq 316$
- $7x_0 \leq 223$
- $15x_0 \leq 263$
- $2x_0 \leq 131$
- $16x_1 \leq 169$
- $17x_1 \leq 316$
- $21x_1 \leq 223$
- $5x_1 \leq 263$
- $4x_1 \leq 131$
- $3x_2 \leq 169$
- $2x_2 \leq 316$
- $13x_2 \leq 223$
- $7x_2 \leq 263$
- $7x_2 \leq 131$
- $16x_0 + 3x_2 \geq 41$
- $21x_1 + 13x_2 \geq 33$
- $7x_0 + 13x_2 \geq 29$
- $7x_0 + 21x_1 + 13x_2 \geq 73$
- $15x_0 + 7x_2 \geq 84$
- $15x_0 + 5x_1 \geq 83$
- $5x_1 + 7x_2 \geq 64$
- $2x_0 + 7x_2 \geq 15$
- $4x_1 + 7x_2 \geq 22$
- $2x_0 + 4x_1 + 7x_2 \geq 29$
- $16x_0 + 16x_1 \leq 72$
- $16x_0 + 3x_2 \leq 152$
- $16x_1 + 3x_2 \leq 110$
- $16x_0 + 16x_1 + 3x_2 \leq 110$
- $17x_1 + 2x_2 \leq 106$
- $11x_0 + 17x_1 + 2x_2 \leq 106$
- $7x_0 + 21x_1 \leq 211$
- $21x_1 + 13x_2 \leq 159$
- $7x_0 + 21x_1 + 13x_2 \leq 159$
- $5x_1 + 7x_2 \leq 210$
- $15x_0 + 5x_1 \leq 202$
- $15x_0 + 5x_1 + 7x_2 \leq 202$
- $4x_1 + 7x_2 \leq 107$
- $2x_0 + 4x_1 + 7x_2 \leq 107$

## Step 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Mary')
    ],
    'objective_function': '2*x0 + 7*x1 + 7*x2',
    'constraints': [
        '16*x0 <= 169',
        '11*x0 <= 316',
        '7*x0 <= 223',
        '15*x0 <= 263',
        '2*x0 <= 131',
        '16*x1 <= 169',
        '17*x1 <= 316',
        '21*x1 <= 223',
        '5*x1 <= 263',
        '4*x1 <= 131',
        '3*x2 <= 169',
        '2*x2 <= 316',
        '13*x2 <= 223',
        '7*x2 <= 263',
        '7*x2 <= 131',
        '16*x0 + 3*x2 >= 41',
        '21*x1 + 13*x2 >= 33',
        '7*x0 + 13*x2 >= 29',
        '7*x0 + 21*x1 + 13*x2 >= 73',
        '15*x0 + 7*x2 >= 84',
        '15*x0 + 5*x1 >= 83',
        '5*x1 + 7*x2 >= 64',
        '2*x0 + 7*x2 >= 15',
        '4*x1 + 7*x2 >= 22',
        '2*x0 + 4*x1 + 7*x2 >= 29',
        '16*x0 + 16*x1 <= 72',
        '16*x0 + 3*x2 <= 152',
        '16*x1 + 3*x2 <= 110',
        '16*x0 + 16*x1 + 3*x2 <= 110',
        '17*x1 + 2*x2 <= 106',
        '11*x0 + 17*x1 + 2*x2 <= 106',
        '7*x0 + 21*x1 <= 211',
        '21*x1 + 13*x2 <= 159',
        '7*x0 + 21*x1 + 13*x2 <= 159',
        '5*x1 + 7*x2 <= 210',
        '15*x0 + 5*x1 <= 202',
        '15*x0 + 5*x1 + 7*x2 <= 202',
        '4*x1 + 7*x2 <= 107',
        '2*x0 + 4*x1 + 7*x2 <= 107'
    ]
}
```

## Step 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name='x0', lb=0)  # hours worked by Hank
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Paul
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Mary

    # Define the objective function
    model.setObjective(2*x0 + 7*x1 + 7*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(16*x0 <= 169)
    model.addConstr(11*x0 <= 316)
    model.addConstr(7*x0 <= 223)
    model.addConstr(15*x0 <= 263)
    model.addConstr(2*x0 <= 131)
    model.addConstr(16*x1 <= 169)
    model.addConstr(17*x1 <= 316)
    model.addConstr(21*x1 <= 223)
    model.addConstr(5*x1 <= 263)
    model.addConstr(4*x1 <= 131)
    model.addConstr(3*x2 <= 169)
    model.addConstr(2*x2 <= 316)
    model.addConstr(13*x2 <= 223)
    model.addConstr(7*x2 <= 263)
    model.addConstr(7*x2 <= 131)
    model.addConstr(16*x0 + 3*x2 >= 41)
    model.addConstr(21*x1 + 13*x2 >= 33)
    model.addConstr(7*x0 + 13*x2 >= 29)
    model.addConstr(7*x0 + 21*x1 + 13*x2 >= 73)
    model.addConstr(15*x0 + 7*x2 >= 84)
    model.addConstr(15*x0 + 5*x1 >= 83)
    model.addConstr(5*x1 + 7*x2 >= 64)
    model.addConstr(2*x0 + 7*x2 >= 15)
    model.addConstr(4*x1 + 7*x2 >= 22)
    model.addConstr(2*x0 + 4*x1 + 7*x2 >= 29)
    model.addConstr(16*x0 + 16*x1 <= 72)
    model.addConstr(16*x0 + 3*x2 <= 152)
    model.addConstr(16*x1 + 3*x2 <= 110)
    model.addConstr(16*x0 + 16*x1 + 3*x2 <= 110)
    model.addConstr(17*x1 + 2*x2 <= 106)
    model.addConstr(11*x0 + 17*x1 + 2*x2 <= 106)
    model.addConstr(7*x0 + 21*x1 <= 211)
    model.addConstr(21*x1 + 13*x2 <= 159)
    model.addConstr(7*x0 + 21*x1 + 13*x2 <= 159)
    model.addConstr(5*x1 + 7*x2 <= 210)
    model.addConstr(15*x0 + 5*x1 <= 202)
    model.addConstr(15*x0 + 5*x1 + 7*x2 <= 202)
    model.addConstr(4*x1 + 7*x2 <= 107)
    model.addConstr(2*x0 + 4*x1 + 7*x2 <= 107)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objval)
        print('Hours worked by Hank: ', x0.varValue)
        print('Hours worked by Paul: ', x1.varValue)
        print('Hours worked by Mary: ', x2.varValue)
    else:
        print('No optimal solution found')

solve_optimization_problem()
```