## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Laura', 'hours worked by Bobby', 'hours worked by Peggy', 'hours worked by Paul', 'hours worked by Mary'] which can be represented symbolically as 'x1', 'x2', 'x3', 'x4', 'x5' respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: $1 \times x1 + 2 \times x2 + 9 \times x3 + 9 \times x4 + 1 \times x5$.

## 3: List the constraints in symbolic notation
The constraints are:
- $21x1 \geq 0$ (Laura's paperwork competence rating, but this seems to be a given attribute rather than a constraint)
- $23x2 \geq 0$ (Bobby's paperwork competence rating, similarly)
- $26x3 \geq 0$ (Peggy's paperwork competence rating)
- $15x4 \geq 0$ (Paul's paperwork competence rating)
- $4x5 \geq 0$ (Mary's paperwork competence rating)
- $21x1 + 23x2 \geq 114$
- $26x3 + 15x4 \geq 89$
- $23x2 + 4x5 \geq 124$
- $15x4 + 4x5 \geq 51$
- $21x1 + 4x5 \geq 146$
- $21x1 + 23x2 + 26x3 + 15x4 + 4x5 \geq 146$
- $2x2 - 4x5 \geq 0$
- $21x1 + 15x4 + 4x5 \leq 577$
- $21x1 + 23x2 + 15x4 \leq 708$

## 4: Create the symbolic representation of the problem
```json
{
'sym_variables': [
    ('x1', 'hours worked by Laura'), 
    ('x2', 'hours worked by Bobby'), 
    ('x3', 'hours worked by Peggy'), 
    ('x4', 'hours worked by Paul'), 
    ('x5', 'hours worked by Mary')
],
'objective_function': '1*x1 + 2*x2 + 9*x3 + 9*x4 + 1*x5',
'constraints': [
    '21*x1 + 23*x2 >= 114',
    '26*x3 + 15*x4 >= 89',
    '23*x2 + 4*x5 >= 124',
    '15*x4 + 4*x5 >= 51',
    '21*x1 + 4*x5 >= 146',
    '21*x1 + 23*x2 + 26*x3 + 15*x4 + 4*x5 >= 146',
    '2*x2 - 4*x5 >= 0',
    '21*x1 + 15*x4 + 4*x5 <= 577',
    '21*x1 + 23*x2 + 15*x4 <= 708'
]
}
```

## Step 5: Write the Gurobi code for the optimization problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Laura', lb=0)
    x2 = model.addVar(name='hours_worked_by_Bobby', lb=0)
    x3 = model.addVar(name='hours_worked_by_Peggy', lb=0)
    x4 = model.addVar(name='hours_worked_by_Paul', lb=0)
    x5 = model.addVar(name='hours_worked_by_Mary', lb=0)

    # Define the objective function
    model.setObjective(1*x1 + 2*x2 + 9*x3 + 9*x4 + 1*x5, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(21*x1 + 23*x2 >= 114)
    model.addConstr(26*x3 + 15*x4 >= 89)
    model.addConstr(23*x2 + 4*x5 >= 124)
    model.addConstr(15*x4 + 4*x5 >= 51)
    model.addConstr(21*x1 + 4*x5 >= 146)
    model.addConstr(21*x1 + 23*x2 + 26*x3 + 15*x4 + 4*x5 >= 146)
    model.addConstr(2*x2 - 4*x5 >= 0)
    model.addConstr(21*x1 + 15*x4 + 4*x5 <= 577)
    model.addConstr(21*x1 + 23*x2 + 15*x4 <= 708)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Laura: ', x1.varValue)
        print('Hours worked by Bobby: ', x2.varValue)
        print('Hours worked by Peggy: ', x3.varValue)
        print('Hours worked by Paul: ', x4.varValue)
        print('Hours worked by Mary: ', x5.varValue)
    else:
        print('No optimal solution found')

solve_optimization_problem()
```