To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of all-inclusive tickets as \(x_1\) and the number of regular tickets as \(x_2\).

The objective is to maximize profit. Given that the profit per all-inclusive ticket is $50 and the profit per regular ticket is $20, the objective function can be represented algebraically as:
\[ \text{Maximize:} \quad 50x_1 + 20x_2 \]

The constraints based on the problem description are:
1. The total number of tickets sold cannot exceed 500 due to capacity constraints: \(x_1 + x_2 \leq 500\)
2. At least 100 all-inclusive tickets must be reserved: \(x_1 \geq 100\)
3. At least 3 times as many people prefer regular tickets over all-inclusive tickets: \(x_2 \geq 3x_1\)

Thus, the symbolic representation of the problem in JSON format is:
```json
{
    'sym_variables': [('x1', 'all-inclusive tickets'), ('x2', 'regular tickets')],
    'objective_function': '50*x1 + 20*x2',
    'constraints': ['x1 + x2 <= 500', 'x1 >= 100', 'x2 >= 3*x1']
}
```

Given this symbolic representation, we can now write the Gurobi code in Python to solve the optimization problem:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name='all_inclusive_tickets', vtype=GRB.INTEGER, lb=0)
x2 = m.addVar(name='regular_tickets', vtype=GRB.INTEGER, lb=0)

# Set the objective function
m.setObjective(50*x1 + 20*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 500, name='total_tickets')
m.addConstr(x1 >= 100, name='all_inclusive_minimum')
m.addConstr(x2 >= 3*x1, name='regular_vs_all_inclusive')

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"All-inclusive tickets: {x1.x}")
    print(f"Regular tickets: {x2.x}")
    print(f"Maximum profit: ${50*x1.x + 20*x2.x}")
else:
    print("No optimal solution found")
```