## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of bottles of perfume.
- $x_2$ represents the number of bottles of mascara.

## Step 2: Convert the natural language description into a symbolic representation
The objective is to maximize profit. The profit from selling $x_1$ bottles of perfume is $20x_1$ and from selling $x_2$ bottles of mascara is $15x_2$. Therefore, the objective function is $20x_1 + 15x_2$.

## 3: Define the constraints
1. Budget constraint: The store has a budget of $20,000. Each bottle of perfume costs $50 and each bottle of mascara costs $40. So, $50x_1 + 40x_2 \leq 20000$.
2. Perfume sales constraint: At least 20 but at most 40 bottles of perfume will be sold. So, $20 \leq x_1 \leq 40$.
3. Mascara sales constraint: The number of mascara sold is at most a third the number of perfume sold. So, $x_2 \leq \frac{1}{3}x_1$.
4. Non-negativity constraint: $x_1 \geq 0$ and $x_2 \geq 0$. However, given the specific bounds on $x_1$, the non-negativity constraint is implicitly satisfied.

## 4: Symbolic representation of the optimization problem
```json
{
'sym_variables': [('x1', 'bottles of perfume'), ('x2', 'bottles of mascara')],
'objective_function': '20*x1 + 15*x2',
'constraints': [
    '50*x1 + 40*x2 <= 20000',
    '20 <= x1 <= 40',
    'x2 <= (1/3)*x1',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 5: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, ub=40, name="perfume")
    x2 = model.addVar(lb=0, name="mascara")

    # Set objective function
    model.setObjective(20*x1 + 15*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(50*x1 + 40*x2 <= 20000, name="budget_constraint")
    model.addConstr(x1 >= 20, name="min_perfume_constraint")
    model.addConstr(x1 <= 40, name="max_perfume_constraint")
    model.addConstr(x2 <= (1/3)*x1, name="mascara_vs_perfume_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of bottles of perfume to buy and sell: {x1.varValue}")
        print(f"Number of bottles of mascara to buy and sell: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```