To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints.

Let's define:
- $x_1$ as the number of scarfs made,
- $x_2$ as the number of toques made.

The objective is to maximize profit, given that each scarf yields a profit of $20 and each toque yields a profit of $25. Therefore, the objective function can be written as:
\[ \text{Maximize: } 20x_1 + 25x_2 \]

Given the constraints:
- The factory has 250,000 knitting minutes available.
- A scarf takes 20 minutes to knit and a toque takes 30 minutes to knit.
- The factory must make at least 5,000 scarfs and at least 3,000 toques.

The constraints can be formulated as follows:
1. Knitting time constraint: $20x_1 + 30x_2 \leq 250,000$
2. Minimum scarf production: $x_1 \geq 5,000$
3. Minimum toque production: $x_2 \geq 3,000$
4. Non-negativity constraints (since the number of items cannot be negative): $x_1 \geq 0$ and $x_2 \geq 0$

However, since $x_1$ and $x_2$ represent quantities of scarfs and toques, which must be integers, we also need to ensure that our solution method accounts for this integrality constraint.

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of scarfs'), ('x2', 'number of toques')],
    'objective_function': '20*x1 + 25*x2',
    'constraints': ['20*x1 + 30*x2 <= 250000', 'x1 >= 5000', 'x2 >= 3000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll need to ensure that the variables are defined as integers. Here's how you can implement it:

```python
from gurobipy import *

# Create a model
m = Model("Knitting_Factory")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="scarfs")
x2 = m.addVar(vtype=GRB.INTEGER, name="toques")

# Set objective function
m.setObjective(20*x1 + 25*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 30*x2 <= 250000, "knitting_time")
m.addConstr(x1 >= 5000, "min_scarfs")
m.addConstr(x2 >= 3000, "min_toques")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of scarfs: {x1.x}")
    print(f"Number of toques: {x2.x}")
    print(f"Maximum profit: ${20*x1.x + 25*x2.x}")
else:
    print("No optimal solution found")
```