## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Paul', 'hours worked by Bill', and 'hours worked by Jean', 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 Paul
- $x_1$ : hours worked by Bill
- $x_2$ : hours worked by Jean

## 3: Define the objective function in symbolic notation
The objective function to minimize is $6x_0 + 4x_1 + 3x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $7x_0 \geq 0$ (not explicitly needed as $x_0$ can be negative but is bounded by other constraints)
- $12x_0 \geq 0$ (similarly, not explicitly needed)
- $4x_1 \geq 0$ (not explicitly needed)
- $2x_1 \geq 0$ (not explicitly needed)
- $3x_2 \geq 0$ (not explicitly needed)
- $12x_2 \geq 0$ (not explicitly needed)
- $7x_0 + 3x_2 \geq 31$
- $7x_0 + 4x_1 \geq 20$
- $7x_0 + 4x_1 + 3x_2 \geq 31$
- $7x_0 + 4x_1 + 3x_2 \geq 31$ (redundant with the previous one)
- $12x_0 + 12x_2 \geq 26$
- $12x_0 + 2x_1 \geq 31$
- $12x_0 + 2x_1 + 12x_2 \geq 31$
- $6x_0 - x_1 \geq 0$
- $4x_1 - 5x_2 \geq 0$
- $7x_0 + 4x_1 + 3x_2 \leq 84$
- $12x_0 + 2x_1 + 12x_2 \leq 123$
- $x_2$ is an integer

## 5: Provide the symbolic representation in the required format
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Paul'), 
        ('x1', 'hours worked by Bill'), 
        ('x2', 'hours worked by Jean')
    ], 
    'objective_function': '6*x0 + 4*x1 + 3*x2', 
    'constraints': [
        '7*x0 + 3*x2 >= 31',
        '7*x0 + 4*x1 >= 20',
        '7*x0 + 4*x1 + 3*x2 >= 31',
        '12*x0 + 12*x2 >= 26',
        '12*x0 + 2*x1 >= 31',
        '12*x0 + 2*x1 + 12*x2 >= 31',
        '6*x0 - x1 >= 0',
        '4*x1 - 5*x2 >= 0',
        '7*x0 + 4*x1 + 3*x2 <= 84',
        '12*x0 + 2*x1 + 12*x2 <= 123',
        'x2 == int(x2)'
    ]
}
```

## Step 6: 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="x0", lb=0)  # hours worked by Paul
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bill
    x2 = model.addVar(name="x2", lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by Jean

    # Define the objective function
    model.setObjective(6 * x0 + 4 * x1 + 3 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(7 * x0 + 3 * x2 >= 31)
    model.addConstr(7 * x0 + 4 * x1 >= 20)
    model.addConstr(7 * x0 + 4 * x1 + 3 * x2 >= 31)
    model.addConstr(12 * x0 + 12 * x2 >= 26)
    model.addConstr(12 * x0 + 2 * x1 >= 31)
    model.addConstr(12 * x0 + 2 * x1 + 12 * x2 >= 31)
    model.addConstr(6 * x0 - x1 >= 0)
    model.addConstr(4 * x1 - 5 * x2 >= 0)
    model.addConstr(7 * x0 + 4 * x1 + 3 * x2 <= 84)
    model.addConstr(12 * x0 + 2 * x1 + 12 * x2 <= 123)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```