## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- $x_1$ : pen testers
- $x_2$ : honeypots
- $x_3$ : system administrators
- $x_4$ : patches per day
- $x_5$ : automatic alerts
- $x_6$ : SOC operators
- $x_7$ : security onions

## Step 2: Formulate the objective function
The objective function to maximize is: $3x_1 + 7x_2 + 3x_3 + 7x_4 + 4x_5 + x_6 + x_7$

## Step 3: List the constraints
Constraints are given in the problem description. For brevity and due to the extensive list, we'll directly incorporate them into the Gurobi model.

## 4: Convert the problem into a symbolic representation
```json
{
'sym_variables': [
    ('x1', 'pen testers'),
    ('x2', 'honeypots'),
    ('x3', 'system administrators'),
    ('x4', 'patches per day'),
    ('x5', 'automatic alerts'),
    ('x6', 'SOC operators'),
    ('x7', 'security onions')
],
'objective_function': '3*x1 + 7*x2 + 3*x3 + 7*x4 + 4*x5 + x6 + x7',
'constraints': [
    '3*x1 + 7*x2 + 18*x3 + 16*x4 + 4*x5 + 11*x6 + 14*x7 <= 225',
    '3*x1 + 7*x2 + 18*x3 + 16*x4 + 4*x5 + 11*x6 + 14*x7 >= 0', # Implicit non-negativity
    '3*x1 >= 20 - 7*x2 - 18*x3 - 16*x4 - 4*x5 - 11*x6 - 14*x7', 
    '7*x2 + 18*x3 + 11*x6 >= 20 - 3*x1 - 16*x4 - 4*x5 - 14*x7',
    # ... (rest of the constraints)
    'x1 + x4 + x7 >= 20/3',
    'x2 + x3 + x6 >= 20/11',
    # ... (rest of the constraints)
    'x1, x2, x3, x4, x5, x6, x7 >= 0', # Implicit non-negativity
    'x1, x2, x3, x4, x5, x6, x7 == int' # Integer constraint
]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="pen_testers", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="honeypots", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="system_administrators", vtype=gurobi.GRB.INTEGER)
    x4 = model.addVar(name="patches_per_day", vtype=gurobi.GRB.INTEGER)
    x5 = model.addVar(name="automatic_alerts", vtype=gurobi.GRB.INTEGER)
    x6 = model.addVar(name="SOC_operators", vtype=gurobi.GRB.INTEGER)
    x7 = model.addVar(name="security_onions", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(3*x1 + 7*x2 + 3*x3 + 7*x4 + 4*x5 + x6 + x7, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(3*x1 + 7*x2 + 18*x3 + 16*x4 + 4*x5 + 11*x6 + 14*x7 <= 225)

    # Network integrity impact constraints
    model.addConstr(3*x1 + 16*x4 + 14*x7 >= 20)
    model.addConstr(7*x2 + 18*x3 + 11*x6 >= 20)
    model.addConstr(7*x2 + 16*x4 + 4*x5 >= 20)
    model.addConstr(3*x1 + 18*x3 + 4*x5 >= 20)
    model.addConstr(3*x1 + 4*x5 + 11*x6 >= 20)
    model.addConstr(16*x4 + 4*x5 + 11*x6 >= 20)
    model.addConstr(16*x4 + 11*x6 + 14*x7 >= 20)
    model.addConstr(7*x2 + 18*x3 + 14*x7 >= 20)
    model.addConstr(3*x1 + 7*x2 + 18*x3 >= 20)
    model.addConstr(18*x3 + 16*x4 + 14*x7 >= 20)
    model.addConstr(3*x1 + 18*x3 + 14*x7 >= 20)
    model.addConstr(3*x1 + 11*x6 + 14*x7 >= 20)
    model.addConstr(3*x1 + 16*x4 + 14*x7 >= 23)
    model.addConstr(7*x2 + 18*x3 + 11*x6 >= 23)
    model.addConstr(7*x2 + 16*x4 + 4*x5 >= 23)
    model.addConstr(3*x1 + 18*x3 + 4*x5 >= 23)
    model.addConstr(3*x1 + 4*x5 + 11*x6 >= 23)
    model.addConstr(16*x4 + 4*x5 + 11*x6 >= 23)
    model.addConstr(16*x4 + 11*x6 + 14*x7 >= 23)
    model.addConstr(7*x2 + 18*x3 + 14*x7 >= 23)
    model.addConstr(3*x1 + 7*x2 + 18*x3 >= 23)
    model.addConstr(18*x3 + 16*x4 + 14*x7 >= 23)
    model.addConstr(3*x1 + 18*x3 + 14*x7 >= 23)
    model.addConstr(3*x1 + 11*x6 + 14*x7 >= 23)

    # Upper bound constraints
    model.addConstr(16*x4 + 14*x7 <= 86)
    model.addConstr(7*x2 + 11*x6 <= 78)
    model.addConstr(7*x2 + 14*x7 <= 198)
    model.addConstr(4*x5 + 11*x6 <= 199)
    model.addConstr(3*x1 + 16*x4 <= 176)
    model.addConstr(3*x1 + 18*x3 <= 140)
    model.addConstr(18*x3 + 4*x5 <= 188)
    model.addConstr(7*x2 + 18*x3 <= 195)
    model.addConstr(18*x3 + 16*x4 + 4*x5 <= 62)
    model.addConstr(3*x1 + 18*x3 + 11*x6 <= 97)
    model.addConstr(3*x1 + 16*x4 + 14*x7 <= 168)
    model.addConstr(3*x1 + 11*x6 + 14*x7 <= 55)
    model.addConstr(18*x3 + 11*x6 + 14*x7 <= 101)
    model.addConstr(16*x4 + 4*x5 + 11*x6 <= 82)
    model.addConstr(3*x1 + 7*x2 + 18*x3 + 16*x4 + 4*x5 + 11*x6 + 14*x7 <= 82)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("pen testers: ", x1.varValue)
        print("honeypots: ", x2.varValue)
        print("system administrators: ", x3.varValue)
        print("patches per day: ", x4.varValue)
        print("automatic alerts: ", x5.varValue)
        print("SOC operators: ", x6.varValue)
        print("security onions: ", x7.varValue)
    else:
        print("No solution found")

optimize_problem()
```