To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of interest (number of stuffed donuts and pastries), formulating the objective function based on these variables, and translating the constraints into mathematical expressions.

Let's denote:
- \(x_1\) as the number of stuffed donuts.
- \(x_2\) as the number of stuffed pastries.

The objective is to maximize profit. Given that each stuffed donut brings a profit of $2 and each pastry brings a profit of $4, the objective function can be written as:
\[ \text{Maximize:} \quad 2x_1 + 4x_2 \]

The constraints are based on the time availability of the baking and stuffing machines. Each stuffed donut requires 2 minutes on the baking machine and 3 minutes on the stuffing machine, while each pastry requires 5 minutes on the baking machine and 2 minutes on the stuffing machine. The total available time for the baking machine is 10,000 minutes, and for the stuffing machine, it's 7,000 minutes. Thus, the constraints can be formulated as:
\[ 2x_1 + 5x_2 \leq 10000 \] (baking machine constraint)
\[ 3x_1 + 2x_2 \leq 7000 \] (stuffing machine constraint)

Additionally, we have non-negativity constraints since the number of donuts and pastries cannot be negative:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Now, let's represent this problem in the requested JSON format:

```json
{
    'sym_variables': [('x1', 'number of stuffed donuts'), ('x2', 'number of stuffed pastries')],
    'objective_function': '2*x1 + 4*x2',
    'constraints': ['2*x1 + 5*x2 <= 10000', '3*x1 + 2*x2 <= 7000', 'x1 >= 0', 'x2 >= 0']
}
```

Next, we will translate this problem into Gurobi code in Python:

```python
from gurobipy import *

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

# Define the variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="stuffed_donuts")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="stuffed_pastries")

# Set the objective function
m.setObjective(2*x1 + 4*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x1 + 5*x2 <= 10000, "baking_machine")
m.addConstr(3*x1 + 2*x2 <= 7000, "stuffing_machine")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of stuffed donuts: {x1.x}")
    print(f"Number of stuffed pastries: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```