## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by Ringo', and 'hours worked by Laura'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_1$ for 'hours worked by Peggy'
- $x_2$ for 'hours worked by Ringo'
- $x_3$ for 'hours worked by Laura'

## 3: Define the objective function in symbolic notation
The objective function to maximize is $6.32x_1 + 6.84x_2 + 4.46x_3$.

## 4: Define the constraints in symbolic notation
The constraints are:
- $1x_1 \leq 179$ (Peggy's organization score constraint, but it's given as a fixed value, so it's not directly a constraint in the traditional sense)
- $12x_2 \leq 179$ (Ringo's organization score constraint, similarly)
- $3x_3 \leq 179$ (Laura's organization score constraint)
- $1x_1 + 12x_2 \leq 60$ (combined organization score constraint for Peggy and Ringo)
- $1x_1 + 3x_3 \leq 147$ (combined organization score constraint for Peggy and Laura)
- $1x_1 + 12x_2 + 3x_3 \leq 147$ (combined organization score constraint for all)

However, given that $x_0=1$, $x_1=12$, and $x_2=3$ are provided for $r_0$, it seems these are coefficients or given values rather than variables to optimize. The actual constraints from the problem statement are:
- $x_1 \geq 0$ (implicit, as hours cannot be negative)
- $x_2 \geq 0$ (implicit)
- $x_3 \geq 0$ (implicit)

## 5: Correctly interpret the given organization scores
Given that Peggy's organization score is 1, Ringo's is 12, and Laura's is 3, these seem to be coefficients of $x_1$, $x_2$, and $x_3$ in the constraints rather than the variables themselves.

## 6: Formulate the optimization problem with correct constraints
The correct constraints based on the problem statement are:
- $1x_1 + 12x_2 \leq 60$
- $1x_1 + 3x_3 \leq 147$
- $1x_1 + 12x_2 + 3x_3 \leq 147$

## 7: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Ringo'), ('x3', 'hours worked by Laura')],
    'objective_function': '6.32*x1 + 6.84*x2 + 4.46*x3',
    'constraints': [
        '1*x1 + 12*x2 <= 60',
        '1*x1 + 3*x3 <= 147',
        '1*x1 + 12*x2 + 3*x3 <= 147',
        'x1 >= 0',
        'x2 >= 0',
        'x3 >= 0'
    ]
}
```

## 8: 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
    x1 = model.addVar(lb=0, name="hours_worked_by_Peggy", vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(lb=0, name="hours_worked_by_Ringo", vtype=gurobi.GRB.CONTINUOUS)
    x3 = model.addVar(lb=0, name="hours_worked_by_Laura", vtype=gurobi.GRB.CONTINUOUS)

    # Define the objective function
    model.setObjective(6.32*x1 + 6.84*x2 + 4.46*x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 + 12*x2 <= 60)
    model.addConstr(x1 + 3*x3 <= 147)
    model.addConstr(x1 + 12*x2 + 3*x3 <= 147)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Peggy: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Hours worked by Laura: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```