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 trains and planes, formulating the objective function based on profits, and translating the constraints into algebraic expressions.

Let's denote:
- $x_1$ as the number of trains produced.
- $x_2$ as the number of planes produced.

The objective is to maximize profit. Given that each train yields a profit of $50 and each plane yields a profit of $60, the objective function can be written as:
\[ \text{Maximize: } 50x_1 + 60x_2 \]

The constraints are as follows:
1. Time constraint for woodworker time: Each train requires 30 minutes, and each plane requires 40 minutes. The total available time is 4000 minutes.
\[ 30x_1 + 40x_2 \leq 4000 \]
2. Production requirement: The store must make at least thrice the number of planes as trains.
\[ x_2 \geq 3x_1 \]

Non-negativity constraints:
- $x_1 \geq 0$ (since we cannot produce a negative number of trains)
- $x_2 \geq 0$ (since we cannot produce a negative number of planes)

Therefore, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of trains'), ('x2', 'number of planes')],
    'objective_function': '50*x1 + 60*x2',
    'constraints': [
        '30*x1 + 40*x2 <= 4000',
        'x2 >= 3*x1',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Toy Store Optimization")

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="trains")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="planes")

# Set the objective function
m.setObjective(50*x1 + 60*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(30*x1 + 40*x2 <= 4000, "time_constraint")
m.addConstr(x2 >= 3*x1, "production_requirement")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of trains: {x1.x}")
    print(f"Number of planes: {x2.x}")
    print(f"Maximum profit: ${50*x1.x + 60*x2.x:.2f}")
else:
    print("No optimal solution found")
```