## Step 1: Define the symbolic representation of the problem
Let's denote the number of Sandwich A to be made as $x_1$ and the number of Sandwich B to be made as $x_2$. The objective is to maximize profit, where the profit per Sandwich A is $5 and per Sandwich B is $6. Therefore, the objective function can be represented as $5x_1 + 6x_2$.

## Step 2: Identify the constraints based on the given resources
The shop has 500 units of cheddar cheese and 400 units of American cheese available. Sandwich A requires 3 units of cheddar cheese and 3 units of American cheese, while Sandwich B requires 5 units of cheddar cheese and 2 units of American cheese. This gives us the following constraints:
- $3x_1 + 5x_2 \leq 500$ (cheddar cheese constraint)
- $3x_1 + 2x_2 \leq 400$ (American cheese constraint)
Additionally, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of sandwiches cannot be negative.

## 3: Symbolic representation in the required format
The symbolic variables are:
- $x_1$ for Sandwich A
- $x_2$ for Sandwich B

The objective function is: $5x_1 + 6x_2$

The constraints are:
- $3x_1 + 5x_2 \leq 500$
- $3x_1 + 2x_2 \leq 400$
- $x_1 \geq 0$
- $x_2 \geq 0$

In the required JSON format:
```json
{
'sym_variables': [('x1', 'Sandwich A'), ('x2', 'Sandwich B')],
'objective_function': '5*x1 + 6*x2',
'constraints': ['3*x1 + 5*x2 <= 500', '3*x1 + 2*x2 <= 400', 'x1 >= 0', 'x2 >= 0']
}
```

## 4: Convert the problem into Gurobi code
Now, let's implement this problem using Gurobi in Python:
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="Sandwich_A", lb=0, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="Sandwich_B", lb=0, ub=gurobi.GRB.INFINITY)

    # Set the objective function
    model.setObjective(5 * x1 + 6 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(3 * x1 + 5 * x2 <= 500, name="cheddar_constraint")
    model.addConstr(3 * x1 + 2 * x2 <= 400, name="american_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Sandwich A = {x1.varValue}, Sandwich B = {x2.varValue}")
        print(f"Maximum profit: ${5 * x1.varValue + 6 * x2.varValue}")
    else:
        print("No optimal solution found")

solve_sandwich_problem()
```