## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale', 'hours worked by Laura', and 'hours worked by Paul', which can be represented symbolically 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 $2x_0 + 4x_1 + 2x_2$.

## Step 3: List all the constraints in symbolic notation
The constraints given are:
- $13x_0 \geq 0$ (not a constraint as it's always true given $x_0 \geq 0$)
- $10x_0 \geq 0$ (not a constraint as it's always true given $x_0 \geq 0$)
- $29x_1 \geq 0$ (not a constraint as it's always true given $x_1 \geq 0$)
- $3x_1 \geq 0$ (not a constraint as it's always true given $x_1 \geq 0$)
- $20x_2 \geq 0$ (not a constraint as it's always true given $x_2 \geq 0$)
- $25x_2 \geq 0$ (not a constraint as it's always true given $x_2 \geq 0$)
- $13x_0 + 29x_1 \geq 107$
- $13x_0 + 29x_1 + 20x_2 \geq 84$
- $13x_0 + 29x_1 + 20x_2 \geq 84$ (redundant with the previous one)
- $10x_0 + 3x_1 \geq 76$
- $10x_0 + 25x_2 \geq 58$
- $10x_0 + 3x_1 + 25x_2 \geq 58$
- $x_1 - 5x_2 \geq 0$
- $13x_0 + 29x_1 \leq 392$
- $13x_0 + 20x_2 \leq 343$
- $x_0, x_1, x_2$ are integers.

## 4: Define the symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x0', 'hours worked by Dale'), ('x1', 'hours worked by Laura'), ('x2', 'hours worked by Paul')],
'objective_function': '2*x0 + 4*x1 + 2*x2',
'constraints': [
    '13*x0 + 29*x1 >= 107',
    '13*x0 + 29*x1 + 20*x2 >= 84',
    '10*x0 + 3*x1 >= 76',
    '10*x0 + 25*x2 >= 58',
    '10*x0 + 3*x1 + 25*x2 >= 58',
    'x1 - 5*x2 >= 0',
    '13*x0 + 29*x1 <= 392',
    '13*x0 + 20*x2 <= 343',
    'x0 >= 0',
    'x1 >= 0',
    'x2 >= 0',
    'x0 == int(x0)',
    'x1 == int(x1)',
    'x2 == int(x2)'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Dale
    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 Paul

    # Objective function
    model.setObjective(2 * x0 + 4 * x1 + 2 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(13 * x0 + 29 * x1 >= 107)
    model.addConstr(13 * x0 + 29 * x1 + 20 * x2 >= 84)
    model.addConstr(10 * x0 + 3 * x1 >= 76)
    model.addConstr(10 * x0 + 25 * x2 >= 58)
    model.addConstr(10 * x0 + 3 * x1 + 25 * x2 >= 58)
    model.addConstr(x1 - 5 * x2 >= 0)
    model.addConstr(13 * x0 + 29 * x1 <= 392)
    model.addConstr(13 * x0 + 20 * x2 <= 343)

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Dale: {x0.varValue}")
        print(f"Hours worked by Laura: {x1.varValue}")
        print(f"Hours worked by Paul: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

optimize_problem()
```