## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables and the objective function, and then list the constraints.

### Variables
- $x_1$: Number of electric cars produced by the electric car factory.
- $x_2$: Number of gas cars produced by the gas car factory.

### Objective Function
The objective is to maximize profit. Given that the profit per electric car is $5000 and per gas car is $3000, the objective function can be written as:
\[ \text{Maximize:} \quad 5000x_1 + 3000x_2 \]

### Constraints
1. The electric car factory can make at most 3 electric cars per day: $x_1 \leq 3$.
2. The gas car factory can make at most 5 gas cars per day: $x_2 \leq 5$.
3. The third factory can process at most 5 cars of either type per day: $x_1 + x_2 \leq 5$.
4. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$ and $x_1, x_2$ are integers.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'electric cars produced'), ('x2', 'gas cars produced')],
    'objective_function': '5000*x1 + 3000*x2',
    'constraints': [
        'x1 <= 3',
        'x2 <= 5',
        'x1 + x2 <= 5',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Electric_and_Gas_Cars_Production")

# Define variables
x1 = model.addVar(lb=0, ub=3, vtype=gp.GRB.INTEGER, name="electric_cars")
x2 = model.addVar(lb=0, ub=5, vtype=gp.GRB.INTEGER, name="gas_cars")

# Objective function: Maximize 5000*x1 + 3000*x2
model.setObjective(5000*x1 + 3000*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(x1 + x2 <= 5, name="total_cars_constraint")

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal production levels: Electric cars = {x1.varValue}, Gas cars = {x2.varValue}")
    print(f"Maximum profit: ${model.objVal}")
else:
    print("The model is infeasible or unbounded.")
```