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

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Jean
- $x_1$ : hours worked by Laura
- $x_2$ : hours worked by Hank

## 3: Define the objective function in symbolic notation
The objective function to maximize is $9x_0 + 7x_1 + 5x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $22x_0 \geq 0$ (implicit, as $x_0$ can be 0)
- $14x_0 \geq 0$ (implicit, as $x_0$ can be 0)
- $20x_1 \geq 0$ (implicit, as $x_1$ can be 0)
- $22x_1 \geq 0$ (implicit, as $x_1$ can be 0)
- $25x_2 \geq 0$ (implicit, as $x_2$ can be 0)
- $23x_2 \geq 0$ (implicit, as $x_2$ can be 0)
- $22x_0 + 20x_1 + 25x_2 \geq 118$
- $22x_1 + 23x_2 \geq 41$
- $14x_0 + 22x_1 \geq 67$
- $14x_0 + 23x_2 \geq 48$
- $14x_0 + 22x_1 + 23x_2 \geq 65$
- $22x_0 + 20x_1 \leq 195$
- $22x_0 + 20x_1 + 25x_2 \leq 423$
- $14x_0 + 23x_2 \leq 152$
- $22x_1 + 23x_2 \leq 164$
- $14x_0 + 22x_1 \leq 143$
- $14x_0 + 22x_1 + 23x_2 \leq 143$

## 5: Provide the symbolic representation in JSON format
```json
{
'sym_variables': [
    ('x0', 'hours worked by Jean'),
    ('x1', 'hours worked by Laura'),
    ('x2', 'hours worked by Hank')
],
'objective_function': '9*x0 + 7*x1 + 5*x2',
'constraints': [
    '22*x0 + 20*x1 + 25*x2 >= 118',
    '22*x1 + 23*x2 >= 41',
    '14*x0 + 22*x1 >= 67',
    '14*x0 + 23*x2 >= 48',
    '14*x0 + 22*x1 + 23*x2 >= 65',
    '22*x0 + 20*x1 <= 195',
    '22*x0 + 20*x1 + 25*x2 <= 423',
    '14*x0 + 23*x2 <= 152',
    '22*x1 + 23*x2 <= 164',
    '14*x0 + 22*x1 <= 143',
    '14*x0 + 22*x1 + 23*x2 <= 143'
]
}
```

## Step 6: Write the Gurobi code
```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 Jean
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Laura
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Hank

    # Define the objective function
    model.setObjective(9 * x0 + 7 * x1 + 5 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(22 * x0 + 20 * x1 + 25 * x2 >= 118)
    model.addConstr(22 * x1 + 23 * x2 >= 41)
    model.addConstr(14 * x0 + 22 * x1 >= 67)
    model.addConstr(14 * x0 + 23 * x2 >= 48)
    model.addConstr(14 * x0 + 22 * x1 + 23 * x2 >= 65)
    model.addConstr(22 * x0 + 20 * x1 <= 195)
    model.addConstr(22 * x0 + 20 * x1 + 25 * x2 <= 423)
    model.addConstr(14 * x0 + 23 * x2 <= 152)
    model.addConstr(22 * x1 + 23 * x2 <= 164)
    model.addConstr(14 * x0 + 22 * x1 <= 143)
    model.addConstr(14 * x0 + 22 * x1 + 23 * x2 <= 143)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objval)
        print('Hours worked by Jean: ', x0.varValue)
        print('Hours worked by Laura: ', x1.varValue)
        print('Hours worked by Hank: ', x2.varValue)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```