## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by Laura', 'hours worked by Ringo', and 'hours worked by Jean', which can be represented symbolically as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $6x_1 + 6x_2 + 8x_3 + 7x_4$.

## Step 3: List the constraints in symbolic notation
The constraints are:
1. $7x_1 \geq 85 - 22x_2$ is not correct, it is $7x_1 + 22x_2 \geq 85$,
2. $7x_1 + 12x_4 \geq 69$,
3. $7x_1 + 28x_3 \geq 119$,
4. $22x_2 + 12x_4 \geq 76$,
5. $22x_2 + 28x_3 \geq 83$,
6. $7x_1 + 22x_2 + 28x_3 \geq 144$,
7. $7x_1 + 28x_3 + 12x_4 \geq 144$,
8. $22x_2 + 28x_3 + 12x_4 \geq 144$,
9. $7x_1 + 22x_2 + 12x_4 \geq 144$,
10. $7x_1 + 22x_2 + 28x_3 \geq 90$,
11. $7x_1 + 28x_3 + 12x_4 \geq 90$,
12. $22x_2 + 28x_3 + 12x_4 \geq 90$,
13. $7x_1 + 22x_2 + 12x_4 \geq 90$,
14. $7x_1 + 22x_2 + 28x_3 \geq 86$,
15. $7x_1 + 28x_3 + 12x_4 \geq 86$,
16. $22x_2 + 28x_3 + 12x_4 \geq 86$,
17. $7x_1 + 22x_2 + 12x_4 \geq 86$,
18. $7x_1 + 22x_2 + 28x_3 \geq 145$,
19. $7x_1 + 28x_3 + 12x_4 \geq 145$,
20. $22x_2 + 28x_3 + 12x_4 \geq 145$,
21. $7x_1 + 22x_2 + 12x_4 \geq 145$,
22. $7x_1 + 22x_2 + 28x_3 + 12x_4 \geq 145$,
23. $-3x_2 + 5x_3 \geq 0$,
24. $7x_3 - 5x_4 \geq 0$,
25. $7x_1 + 12x_4 \leq 269$,
26. $22x_2 + 28x_3 \leq 322$,
27. $28x_3 + 12x_4 \leq 389$,
28. $7x_1 + 28x_3 + 12x_4 \leq 496$.

## 4: Define variable bounds
- $x_1$ is a continuous variable,
- $x_2$ is an integer variable,
- $x_3$ is a continuous variable,
- $x_4$ is an integer variable.

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

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Mary')
x2 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype='I', name='hours_worked_by_Laura')
x3 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Ringo')
x4 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype='I', name='hours_worked_by_Jean')

# Objective function
m.setObjective(6 * x1 + 6 * x2 + 8 * x3 + 7 * x4, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(7 * x1 + 22 * x2 >= 85)
m.addConstr(7 * x1 + 12 * x4 >= 69)
m.addConstr(7 * x1 + 28 * x3 >= 119)
m.addConstr(22 * x2 + 12 * x4 >= 76)
m.addConstr(22 * x2 + 28 * x3 >= 83)
m.addConstr(7 * x1 + 22 * x2 + 28 * x3 >= 144)
m.addConstr(7 * x1 + 28 * x3 + 12 * x4 >= 144)
m.addConstr(22 * x2 + 28 * x3 + 12 * x4 >= 144)
m.addConstr(7 * x1 + 22 * x2 + 12 * x4 >= 144)
m.addConstr(7 * x1 + 22 * x2 + 28 * x3 >= 90)
m.addConstr(7 * x1 + 28 * x3 + 12 * x4 >= 90)
m.addConstr(22 * x2 + 28 * x3 + 12 * x4 >= 90)
m.addConstr(7 * x1 + 22 * x2 + 12 * x4 >= 90)
m.addConstr(7 * x1 + 22 * x2 + 28 * x3 >= 86)
m.addConstr(7 * x1 + 28 * x3 + 12 * x4 >= 86)
m.addConstr(22 * x2 + 28 * x3 + 12 * x4 >= 86)
m.addConstr(7 * x1 + 22 * x2 + 12 * x4 >= 86)
m.addConstr(7 * x1 + 22 * x2 + 28 * x3 >= 145)
m.addConstr(7 * x1 + 28 * x3 + 12 * x4 >= 145)
m.addConstr(22 * x2 + 28 * x3 + 12 * x4 >= 145)
m.addConstr(7 * x1 + 22 * x2 + 12 * x4 >= 145)
m.addConstr(7 * x1 + 22 * x2 + 28 * x3 + 12 * x4 >= 145)
m.addConstr(-3 * x2 + 5 * x3 >= 0)
m.addConstr(7 * x3 - 5 * x4 >= 0)
m.addConstr(7 * x1 + 12 * x4 <= 269)
m.addConstr(22 * x2 + 28 * x3 <= 322)
m.addConstr(28 * x3 + 12 * x4 <= 389)
m.addConstr(7 * x1 + 28 * x3 + 12 * x4 <= 496)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objval)
    print('Hours worked by Mary: ', x1.varValue)
    print('Hours worked by Laura: ', x2.varValue)
    print('Hours worked by Ringo: ', x3.varValue)
    print('Hours worked by Jean: ', x4.varValue)
else:
    print('No solution found')
```

## 6: Symbolic representation
```json
{
    'sym_variables': [
        ['x1', 'hours worked by Mary'],
        ['x2', 'hours worked by Laura'],
        ['x3', 'hours worked by Ringo'],
        ['x4', 'hours worked by Jean']
    ],
    'objective_function': '6*x1 + 6*x2 + 8*x3 + 7*x4',
    'constraints': [
        '7*x1 + 22*x2 >= 85',
        '7*x1 + 12*x4 >= 69',
        '7*x1 + 28*x3 >= 119',
        '22*x2 + 12*x4 >= 76',
        '22*x2 + 28*x3 >= 83',
        '7*x1 + 22*x2 + 28*x3 >= 144',
        '7*x1 + 28*x3 + 12*x4 >= 144',
        '22*x2 + 28*x3 + 12*x4 >= 144',
        '7*x1 + 22*x2 + 12*x4 >= 144',
        '7*x1 + 22*x2 + 28*x3 >= 90',
        '7*x1 + 28*x3 + 12*x4 >= 90',
        '22*x2 + 28*x3 + 12*x4 >= 90',
        '7*x1 + 22*x2 + 12*x4 >= 90',
        '7*x1 + 22*x2 + 28*x3 >= 86',
        '7*x1 + 28*x3 + 12*x4 >= 86',
        '22*x2 + 28*x3 + 12*x4 >= 86',
        '7*x1 + 22*x2 + 12*x4 >= 86',
        '7*x1 + 22*x2 + 28*x3 >= 145',
        '7*x1 + 28*x3 + 12*x4 >= 145',
        '22*x2 + 28*x3 + 12*x4 >= 145',
        '7*x1 + 22*x2 + 12*x4 >= 145',
        '7*x1 + 22*x2 + 28*x3 + 12*x4 >= 145',
        '-3*x2 + 5*x3 >= 0',
        '7*x3 - 5*x4 >= 0',
        '7*x1 + 12*x4 <= 269',
        '22*x2 + 28*x3 <= 322',
        '28*x3 + 12*x4 <= 389',
        '7*x1 + 28*x3 + 12*x4 <= 496'
    ]
}
```