To solve this optimization problem, we first need to translate the natural language description into a symbolic representation. Let's define:

- $x_1$ as the number of bookcases made,
- $x_2$ as the number of garden chairs made.

The objective function is to maximize profit, which can be represented algebraically as:
\[ 270x_1 + 350x_2 \]

Given the constraints:
1. Each bookcase requires 3 hours of woodworking and each garden chair requires 2 hours, with a total of 80 hours available.
2. Each bookcase requires 2 boxes of nails and each garden chair requires 4 boxes, with a total of 70 boxes available.
3. Each bookcase requires 3 units of varnish and each garden chair also requires 3 units, with a total of 90 units available.

The constraints can be represented as:
1. $3x_1 + 2x_2 \leq 80$ (woodworking hours constraint)
2. $2x_1 + 4x_2 \leq 70$ (nails boxes constraint)
3. $3x_1 + 3x_2 \leq 90$ (varnish units constraint)

Also, since the number of bookcases and garden chairs cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of bookcases'), ('x2', 'number of garden chairs')],
    'objective_function': '270*x1 + 350*x2',
    'constraints': ['3*x1 + 2*x2 <= 80', '2*x1 + 4*x2 <= 70', '3*x1 + 3*x2 <= 90', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Nu_Designs_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="bookcases", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="garden_chairs", lb=0)

# Set the objective function
m.setObjective(270*x1 + 350*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x1 + 2*x2 <= 80, "woodworking_hours")
m.addConstr(2*x1 + 4*x2 <= 70, "nails_boxes")
m.addConstr(3*x1 + 3*x2 <= 90, "varnish_units")

# Optimize the model
m.optimize()

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