## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bobby', 'hours worked by Paul', 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 $4x_0 + 5x_1 + 3x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $5x_0 \geq 0$ and $x_0 \leq 70$ (but $x_0$ is bounded by integer and non-negativity implicitly)
- $4x_0 \geq 0$ and $x_0 \leq 68$ (but $x_0$ is bounded by integer and non-negativity implicitly)
- $1x_1 \geq 0$ and $x_1 \leq 70$ (but $x_1$ is bounded by integer and non-negativity implicitly)
- $9x_1 \geq 0$ and $x_1 \leq 68$ (but $x_1$ is bounded by integer and non-negativity implicitly)
- $7x_2 \geq 0$ and $x_2 \leq 70$ (but $x_2$ is bounded by integer and non-negativity implicitly)
- $5x_2 \geq 0$ and $x_2 \leq 68$ (but $x_2$ is bounded by integer and non-negativity implicitly)
- $5x_0 + 7x_2 \geq 16$
- $9x_1 + 5x_2 \geq 18$
- $4x_0 + 9x_1 \geq 12$
- $4x_0 + 9x_1 + 5x_2 \geq 15$
- $5x_0 + 1x_1 \leq 45$
- $1x_1 + 7x_2 \leq 47$
- $5x_0 + 1x_1 + 7x_2 \leq 26$
- $5x_0 + 1x_1 + 7x_2 \leq 26$ (Redundant with previous)
- $4x_0 + 9x_1 \leq 52$
- $9x_1 + 5x_2 \leq 37$
- $4x_0 + 9x_1 + 5x_2 \leq 37$
- $x_0, x_1, x_2$ are integers.

## 4: Write down the problem in a structured format
```json
{
    'sym_variables': [('x0', 'hours worked by Bobby'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Ringo')],
    'objective_function': '4*x0 + 5*x1 + 3*x2',
    'constraints': [
        '5*x0 + 7*x2 >= 16',
        '9*x1 + 5*x2 >= 18',
        '4*x0 + 9*x1 >= 12',
        '4*x0 + 9*x1 + 5*x2 >= 15',
        '5*x0 + x1 <= 45',
        'x1 + 7*x2 <= 47',
        '5*x0 + x1 + 7*x2 <= 26',
        '4*x0 + 9*x1 <= 52',
        '9*x1 + 5*x2 <= 37',
        '4*x0 + 9*x1 + 5*x2 <= 37',
        'x0, x1, x2 >= 0 and are integers'
    ]
}
```

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

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

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

    # Objective function: maximize 4*x0 + 5*x1 + 3*x2
    model.setObjective(4*x0 + 5*x1 + 3*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5*x0 + 7*x2 >= 16)
    model.addConstr(9*x1 + 5*x2 >= 18)
    model.addConstr(4*x0 + 9*x1 >= 12)
    model.addConstr(4*x0 + 9*x1 + 5*x2 >= 15)
    model.addConstr(5*x0 + x1 <= 45)
    model.addConstr(x1 + 7*x2 <= 47)
    model.addConstr(5*x0 + x1 + 7*x2 <= 26)
    model.addConstr(4*x0 + 9*x1 <= 52)
    model.addConstr(9*x1 + 5*x2 <= 37)
    model.addConstr(4*x0 + 9*x1 + 5*x2 <= 37)

    # Non-negativity constraints
    model.addConstr(x0 >= 0)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```