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

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $4x_0 + 6x_1 + 2x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $3x_0 \leq 270$
- $6x_0 \leq 204$
- $21x_0 \leq 197$
- $21x_0 \leq 198$
- $15x_1 \leq 270$
- $17x_1 \leq 204$
- $19x_1 \leq 197$
- $5x_1 \leq 198$
- $16x_2 \leq 270$
- $10x_2 \leq 204$
- $7x_2 \leq 197$
- $7x_2 \leq 198$
- $3x_0 + 16x_2 \leq 101$
- $15x_1 + 16x_2 \leq 204$
- $3x_0 + 15x_1 + 16x_2 \leq 204$
- $17x_1 + 10x_2 \leq 143$
- $6x_0 + 17x_1 \leq 128$
- $6x_0 + 17x_1 + 10x_2 \leq 191$
- $21x_0 + 7x_2 \leq 132$
- $21x_0 + 19x_1 \leq 160$
- $21x_0 + 19x_1 + 7x_2 \leq 160$
- $21x_0 + 7x_2 \leq 69$
- $21x_0 + 5x_1 \leq 192$
- $21x_0 + 5x_1 + 7x_2 \leq 192$

## Step 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'), 
        ('x1', 'hours worked by Jean'), 
        ('x2', 'hours worked by Ringo')
    ], 
    'objective_function': '4*x0 + 6*x1 + 2*x2', 
    'constraints': [
        '3*x0 <= 270', '6*x0 <= 204', '21*x0 <= 197', '21*x0 <= 198',
        '15*x1 <= 270', '17*x1 <= 204', '19*x1 <= 197', '5*x1 <= 198',
        '16*x2 <= 270', '10*x2 <= 204', '7*x2 <= 197', '7*x2 <= 198',
        '3*x0 + 16*x2 <= 101', '15*x1 + 16*x2 <= 204', '3*x0 + 15*x1 + 16*x2 <= 204',
        '17*x1 + 10*x2 <= 143', '6*x0 + 17*x1 <= 128', '6*x0 + 17*x1 + 10*x2 <= 191',
        '21*x0 + 7*x2 <= 132', '21*x0 + 19*x1 <= 160', '21*x0 + 19*x1 + 7*x2 <= 160',
        '21*x0 + 7*x2 <= 69', '21*x0 + 5*x1 <= 192', '21*x0 + 5*x1 + 7*x2 <= 192'
    ]
}
```

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

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Laura", lb=0)
    x1 = model.addVar(name="hours_worked_by_Jean", lb=0)
    x2 = model.addVar(name="hours_worked_by_Ringo", lb=0)

    # Define the objective function
    model.setObjective(4*x0 + 6*x1 + 2*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(3*x0 <= 270)
    model.addConstr(6*x0 <= 204)
    model.addConstr(21*x0 <= 197)
    model.addConstr(21*x0 <= 198)
    model.addConstr(15*x1 <= 270)
    model.addConstr(17*x1 <= 204)
    model.addConstr(19*x1 <= 197)
    model.addConstr(5*x1 <= 198)
    model.addConstr(16*x2 <= 270)
    model.addConstr(10*x2 <= 204)
    model.addConstr(7*x2 <= 197)
    model.addConstr(7*x2 <= 198)
    model.addConstr(3*x0 + 16*x2 <= 101)
    model.addConstr(15*x1 + 16*x2 <= 204)
    model.addConstr(3*x0 + 15*x1 + 16*x2 <= 204)
    model.addConstr(17*x1 + 10*x2 <= 143)
    model.addConstr(6*x0 + 17*x1 <= 128)
    model.addConstr(6*x0 + 17*x1 + 10*x2 <= 191)
    model.addConstr(21*x0 + 7*x2 <= 132)
    model.addConstr(21*x0 + 19*x1 <= 160)
    model.addConstr(21*x0 + 19*x1 + 7*x2 <= 160)
    model.addConstr(21*x0 + 7*x2 <= 69)
    model.addConstr(21*x0 + 5*x1 <= 192)
    model.addConstr(21*x0 + 5*x1 + 7*x2 <= 192)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```