To solve this optimization problem, we first need to define the symbolic representation of the variables and the objective function, along with the constraints.

Let's denote:
- \(x_1\) as the number of small shelves bought.
- \(x_2\) as the number of large shelves bought.

The objective function aims to maximize the total number of action figures stored. Given that a small shelf can hold 20 action figures and a large shelf can hold 30 action figures, the objective function can be represented algebraically as:
\[ \text{Maximize: } 20x_1 + 30x_2 \]

The constraints are based on the available space and budget:
1. Space constraint: Each small shelf takes 3 sq ft of space, and each large shelf takes 6 sq ft. The total available space is 100 sq ft.
\[ 3x_1 + 6x_2 \leq 100 \]
2. Budget constraint: A small shelf costs $50, a large shelf costs $80, and the total budget is $1250.
\[ 50x_1 + 80x_2 \leq 1250 \]
3. Non-negativity constraints: The number of shelves cannot be negative.
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Given this, the symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of small shelves'), ('x2', 'number of large shelves')],
    'objective_function': '20*x1 + 30*x2',
    'constraints': [
        '3*x1 + 6*x2 <= 100',
        '50*x1 + 80*x2 <= 1250',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

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

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

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

# Set the objective function
m.setObjective(20*x1 + 30*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x1 + 6*x2 <= 100, "space_constraint")
m.addConstr(50*x1 + 80*x2 <= 1250, "budget_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Small shelves: {x1.x}")
    print(f"Large shelves: {x2.x}")
    print(f"Total action figures stored: {20*x1.x + 30*x2.x}")
else:
    print("No optimal solution found")
```