## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of laptops sold
- $x_2$ represents the number of desktops sold

## Step 2: Translate the objective function into symbolic notation
The profit per laptop sold is $210, and the profit per desktop sold is $160. The objective function to maximize profit is:
\[ \text{Maximize:} \quad 210x_1 + 160x_2 \]

## 3: Translate the constraints into symbolic notation
The constraints based on the problem description are:
1. The store can spend at most $20,000 on computers, with each laptop costing $500 and each desktop costing $400:
\[ 500x_1 + 400x_2 \leq 20000 \]
2. At least 20 laptops but at most 35 are sold each month:
\[ 20 \leq x_1 \leq 35 \]
3. The number of desktops sold is at most a third of the number of laptops sold:
\[ x_2 \leq \frac{1}{3}x_1 \]
4. Non-negativity constraints (number of items sold cannot be negative):
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
However, given $x_1 \geq 20$, the $x_1 \geq 0$ constraint is implicitly satisfied.

## 4: Compile the symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'laptops'), ('x2', 'desktops')],
'objective_function': '210*x1 + 160*x2',
'constraints': [
    '500*x1 + 400*x2 <= 20000',
    '20 <= x1 <= 35',
    'x2 <= (1/3)*x1',
    'x1 >= 0', 'x2 >= 0'
]
}
```

## 5: Convert to Gurobi code in Python
Now, let's convert this into Gurobi code in Python:
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=20, ub=35, name="laptops")  # At least 20, at most 35 laptops
    x2 = model.addVar(lb=0, name="desktops")  # Non-negative number of desktops

    # Objective function: Maximize 210*x1 + 160*x2
    model.setObjective(210*x1 + 160*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(500*x1 + 400*x2 <= 20000, name="budget_constraint")  # Budget constraint
    model.addConstr(x2 <= (1/3)*x1, name="laptop_desktop_ratio")  # Desktop ratio constraint

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Laptops to sell: {x1.varValue}")
        print(f"Desktops to sell: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```