To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of small and large chessboards to be made, formulating the objective function that represents the total profit, and listing the constraints based on the available time for cutting and assembly.

Let's define:
- $x_1$ as the number of small chessboards
- $x_2$ as the number of large chessboards

The objective function, which is to maximize profit, can be represented as $4x_1 + 8x_2$, since each small chessboard brings a profit of $4 and each large chessboard brings a profit of $8.

The constraints are based on the time available for cutting and assembly:
- Cutting time: $5x_1 + 10x_2 \leq 400$
- Assembly time: $10x_1 + 20x_2 \leq 700$

Additionally, we have non-negativity constraints since the number of chessboards cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:

```json
{
    'sym_variables': [('x1', 'number of small chessboards'), ('x2', 'number of large chessboards')],
    'objective_function': '4*x1 + 8*x2',
    'constraints': ['5*x1 + 10*x2 <= 400', '10*x1 + 20*x2 <= 700', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we will use the following code:

```python
from gurobipy import *

# Create a new model
model = Model("Chessboard_Production")

# Define variables
x1 = model.addVar(name='x1', vtype=GRB.CONTINUOUS, lb=0)
x2 = model.addVar(name='x2', vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
model.setObjective(4*x1 + 8*x2, GRB.MAXIMIZE)

# Add constraints
model.addConstr(5*x1 + 10*x2 <= 400, name='cutting_time')
model.addConstr(10*x1 + 20*x2 <= 700, name='assembly_time')

# Optimize the model
model.optimize()

# Print the solution
if model.status == GRB.OPTIMAL:
    print(f"Optimal solution: x1 = {x1.x}, x2 = {x2.x}")
    print(f"Maximum profit: {model.objVal}")
else:
    print("No optimal solution found")
```