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

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

## 4: List the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$ and $x_0$ is an integer (implicitly from the problem statement that hours worked by Peggy must be a whole number)
- $x_1 \geq 0$ (no restriction on integrality)
- $x_2 \geq 0$ and $x_2$ is an integer (implicitly from the problem statement that hours worked by Hank must be an integer)
- $1x_0 + 1x_1 + 8x_2 \geq 12$ (total combined productivity rating)
- $3x_0 + 2x_2 \leq 27$ (total combined paperwork competence rating from Peggy and Hank)
- $7x_1 + 2x_2 \leq 27$ (total combined paperwork competence rating from Paul and Hank)
- $3x_0 + 7x_1 \leq 34$ (total combined paperwork competence rating from Peggy and Paul)
- $3x_0 + 7x_1 + 2x_2 \leq 34$ (total combined paperwork competence rating from all)
- $1x_1 + 8x_2 \leq 29$ (total combined productivity rating from Paul and Hank)
- $1x_0 + 1x_1 \leq 33$ (total combined productivity rating from Peggy and Paul)
- $1x_0 + 1x_1 + 8x_2 \leq 33$ (total combined productivity rating from all)

## 5: Create the Gurobi model and variables
We will use Gurobi to solve this optimization problem.

## 6: Implement the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="x0", vtype=gp.GRB.INTEGER, lb=0)  # hours worked by Peggy
x1 = m.addVar(name="x1", vtype=gp.GRB.CONTINUOUS, lb=0)  # hours worked by Paul
x2 = m.addVar(name="x2", vtype=gp.GRB.INTEGER, lb=0)  # hours worked by Hank

# Define the objective function
m.setObjective(7*x0 + 3*x1 + 3*x2, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(x0 + x1 + 8*x2 >= 12)  # productivity rating
m.addConstr(3*x0 + 2*x2 <= 27)  # paperwork competence rating (Peggy, Hank)
m.addConstr(7*x1 + 2*x2 <= 27)  # paperwork competence rating (Paul, Hank)
m.addConstr(3*x0 + 7*x1 <= 34)  # paperwork competence rating (Peggy, Paul)
m.addConstr(3*x0 + 7*x1 + 2*x2 <= 34)  # paperwork competence rating (all)
m.addConstr(x1 + 8*x2 <= 29)  # productivity rating (Paul, Hank)
m.addConstr(x0 + x1 <= 33)  # productivity rating (Peggy, Paul)
m.addConstr(x0 + x1 + 8*x2 <= 33)  # productivity rating (all)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Peggy: {x0.varValue}")
    print(f"Hours worked by Paul: {x1.varValue}")
    print(f"Hours worked by Hank: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 7: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Hank')
    ],
    'objective_function': '7*x0 + 3*x1 + 3*x2',
    'constraints': [
        'x0 + x1 + 8*x2 >= 12',
        '3*x0 + 2*x2 <= 27',
        '7*x1 + 2*x2 <= 27',
        '3*x0 + 7*x1 <= 34',
        '3*x0 + 7*x1 + 2*x2 <= 34',
        'x1 + 8*x2 <= 29',
        'x0 + x1 <= 33',
        'x0 + x1 + 8*x2 <= 33'
    ]
}
```