## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of sofas produced
- $x_2$ represents the number of kitchen cabinets produced

## Step 2: Translate the objective function into symbolic notation
The objective function is to maximize profit. Given that each sofa yields a profit of $400 and each kitchen cabinet yields a profit of $1200, the objective function can be written as:
\[ \text{Maximize:} \quad 400x_1 + 1200x_2 \]

## 3: Define the constraints in symbolic notation
The constraints are:
- Lacquer constraint: $3x_1 + 10x_2 \leq 100$ (since 100 gallons of lacquer are available)
- Oak constraint: $10x_1 + 24x_2 \leq 300$ (since 300 lengths of high-quality oak are available)
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$ (since production cannot be negative)

## 4: Compile the symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'sofas'), ('x2', 'kitchen cabinets')],
'objective_function': '400*x1 + 1200*x2',
'constraints': [
    '3*x1 + 10*x2 <= 100',
    '10*x1 + 24*x2 <= 300',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

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

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

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

    # Define the objective function
    model.setObjective(400 * x1 + 1200 * x2, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(3 * x1 + 10 * x2 <= 100, name="lacquer_constraint")
    model.addConstr(10 * x1 + 24 * x2 <= 300, name="oak_constraint")

    # Optimize the model
    model.optimize()

    # Print the results
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal production plan:")
        print(f"Sofas: {x1.varValue}")
        print(f"Kitchen cabinets: {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("The model is infeasible")

east_oak_designs_optimization()
```