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

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is $6x_0 + 7x_1 + 3x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $13x_0 \geq 13$ (Laura's productivity rating constraint, but this seems to be an attribute, not a constraint in the traditional sense. We will consider the given attributes as fixed values.)
- $1x_0 \geq 1$ (Laura's organization score constraint, similarly, this is an attribute)
- $15x_1 \geq 15$ (Bill's productivity rating constraint)
- $3x_1 \geq 3$ (Bill's organization score constraint)
- $16x_2 \geq 16$ (Mary's productivity rating constraint)
- $11x_2 \geq 11$ (Mary's organization score constraint)
- $13x_0 + 15x_1 \geq 20$ (combined productivity rating constraint for Laura and Bill)
- $13x_0 + 15x_1 + 16x_2 \geq 20$ (combined productivity rating constraint for all)
- $1x_0 + 11x_2 \geq 34$ (combined organization score constraint for Laura and Mary)
- $1x_0 + 3x_1 \geq 37$ (combined organization score constraint for Laura and Bill)
- $1x_0 + 3x_1 + 11x_2 \geq 37$ (combined organization score constraint for all)
- $15x_1 + 16x_2 \leq 47$ (combined productivity rating constraint for Bill and Mary)
- $1x_0 + 3x_1 + 11x_2 \leq 134$ (combined organization score constraint for all)

## 5: Correct the understanding of the problem
Upon closer inspection, the "productivity rating" and "organization score" for each individual seem to be attributes rather than constraints. The actual constraints come from the requirements:
- Minimize $6x_0 + 7x_1 + 3x_2$
- Subject to:
  - $13x_0 + 15x_1 \geq 20$
  - $13x_0 + 15x_1 + 16x_2 \geq 20$
  - $x_0 + 11x_2 \geq 34$
  - $x_0 + 3x_1 \geq 37$
  - $x_0 + 3x_1 + 11x_2 \geq 37$
  - $15x_1 + 16x_2 \leq 47$
  - $x_0 + 3x_1 + 11x_2 \leq 134$

## 6: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'hours worked by Laura'), ('x1', 'hours worked by Bill'), ('x2', 'hours worked by Mary')],
'objective_function': '6*x0 + 7*x1 + 3*x2',
'constraints': [
    '13*x0 + 15*x1 >= 20',
    '13*x0 + 15*x1 + 16*x2 >= 20',
    'x0 + 11*x2 >= 34',
    'x0 + 3*x1 >= 37',
    'x0 + 3*x1 + 11*x2 >= 37',
    '15*x1 + 16*x2 <= 47',
    'x0 + 3*x1 + 11*x2 <= 134'
]
}
```

## 7: Write the Gurobi code
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Laura
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bill
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Mary

    # Define the objective function
    model.setObjective(6 * x0 + 7 * x1 + 3 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(13 * x0 + 15 * x1 >= 20)
    model.addConstr(13 * x0 + 15 * x1 + 16 * x2 >= 20)
    model.addConstr(x0 + 11 * x2 >= 34)
    model.addConstr(x0 + 3 * x1 >= 37)
    model.addConstr(x0 + 3 * x1 + 11 * x2 >= 37)
    model.addConstr(15 * x1 + 16 * x2 <= 47)
    model.addConstr(x0 + 3 * x1 + 11 * x2 <= 134)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```