To solve the given problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints in terms of these variables.

Let's define:
- \(x_1\) as the number of waiters.
- \(x_2\) as the number of cooks.

The objective is to minimize the total weekly wage bill, which can be represented as:
\[ \text{Minimize} \quad 147x_1 + 290x_2 \]

The constraints are:
1. The total weekly wage bill should not exceed $17,600:
\[ 147x_1 + 290x_2 \leq 17600 \]
2. The minimum number of staff required is 50:
\[ x_1 + x_2 \geq 50 \]
3. At least 12 staff must be cooks:
\[ x_2 \geq 12 \]
4. The number of cooks should be at least one third the number of waiters (union regulations):
\[ x_2 \geq \frac{1}{3}x_1 \]

All variables \(x_1, x_2\) should be non-negative since they represent numbers of people.

In symbolic notation with natural language objects:
```json
{
  'sym_variables': [('x1', 'number of waiters'), ('x2', 'number of cooks')],
  'objective_function': 'Minimize 147*x1 + 290*x2',
  'constraints': [
    '147*x1 + 290*x2 <= 17600',
    'x1 + x2 >= 50',
    'x2 >= 12',
    'x2 >= (1/3)*x1',
    'x1 >= 0',
    'x2 >= 0'
  ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(147*x1 + 290*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(147*x1 + 290*x2 <= 17600, "wage_bill")
m.addConstr(x1 + x2 >= 50, "total_staff")
m.addConstr(x2 >= 12, "min_cooks")
m.addConstr(x2 >= (1/3)*x1, "union_regulations")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of waiters: {x1.x}")
    print(f"Number of cooks: {x2.x}")
    print(f"Total wage bill: ${147*x1.x + 290*x2.x:.2f}")
else:
    print("No optimal solution found")
```