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

## 3: Define the objective function in symbolic notation
The objective function to minimize is $4x_0 + 7x_1 + 3x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $14x_0 \leq 142$ (computer competence rating for Mary)
- $9x_0 \leq 67$ (organization score for Mary)
- $3x_1 \leq 142$ (computer competence rating for Paul)
- $9x_1 \leq 67$ (organization score for Paul)
- $8x_2 \leq 142$ (computer competence rating for John)
- $9x_2 \leq 67$ (organization score for John)
- $14x_0 + 8x_2 \geq 22$ (total combined computer competence rating from Mary and John)
- $14x_0 + 3x_1 \geq 16$ (total combined computer competence rating from Mary and Paul)
- $14x_0 + 3x_1 + 8x_2 \geq 16$ (total combined computer competence rating from Mary, Paul, and John)
- $9x_0 + 9x_1 \geq 14$ (total combined organization score from Mary and Paul)
- $9x_0 + 9x_2 \geq 8$ (total combined organization score from Mary and John)
- $9x_1 + 9x_2 \geq 21$ (total combined organization score from Paul and John)
- $9x_0 + 9x_1 + 9x_2 \geq 21$ (total combined organization score from Mary, Paul, and John)
- $5x_0 - 8x_2 \geq 0$ (constraint involving Mary and John)
- $x_0 \in \mathbb{Z}$ (integer constraint for Mary)
- $x_1 \in \mathbb{Z}$ (integer constraint for Paul)
- $x_2 \in \mathbb{Z}$ (integer constraint for John)

## 5: Provide the symbolic representation in JSON format
```json
{
'sym_variables': [('x0', 'hours worked by Mary'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by John')],
'objective_function': '4*x0 + 7*x1 + 3*x2',
'constraints': [
'14*x0 <= 142',
'9*x0 <= 67',
'3*x1 <= 142',
'9*x1 <= 67',
'8*x2 <= 142',
'9*x2 <= 67',
'14*x0 + 8*x2 >= 22',
'14*x0 + 3*x1 >= 16',
'14*x0 + 3*x1 + 8*x2 >= 16',
'9*x0 + 9*x1 >= 14',
'9*x0 + 9*x2 >= 8',
'9*x1 + 9*x2 >= 21',
'9*x0 + 9*x1 + 9*x2 >= 21',
'5*x0 - 8*x2 >= 0',
'x0 ∈ ℤ',
'x1 ∈ ℤ',
'x2 ∈ ℤ'
]
}
```

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

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

    # Define the variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Mary
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Paul
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by John

    # Define the objective function
    model.setObjective(4 * x0 + 7 * x1 + 3 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(14 * x0 <= 142)
    model.addConstr(9 * x0 <= 67)
    model.addConstr(3 * x1 <= 142)
    model.addConstr(9 * x1 <= 67)
    model.addConstr(8 * x2 <= 142)
    model.addConstr(9 * x2 <= 67)
    model.addConstr(14 * x0 + 8 * x2 >= 22)
    model.addConstr(14 * x0 + 3 * x1 >= 16)
    model.addConstr(14 * x0 + 3 * x1 + 8 * x2 >= 16)
    model.addConstr(9 * x0 + 9 * x1 >= 14)
    model.addConstr(9 * x0 + 9 * x2 >= 8)
    model.addConstr(9 * x1 + 9 * x2 >= 21)
    model.addConstr(9 * x0 + 9 * x1 + 9 * x2 >= 21)
    model.addConstr(5 * x0 - 8 * x2 >= 0)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Mary: {x0.varValue}")
        print(f"Hours worked by Paul: {x1.varValue}")
        print(f"Hours worked by John: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```